Let U={10,11,12,13,14,15,16,17},A={10,12,14,16},B={11,12,15,16}U=\left\lbrace10,11,12,13,14,15,16,17\right\rbrace,A={\left\lbrace10,12,14,16\right\rbrace},B=\left\lbrace11,12,15,16\right\rbrace. Find:A∩B=A∩B=

A∩B={12,16}A\cap B=\left\lbrace12,16\right\rbrace

A∩B={13,17}A\cap B=\left\lbrace13,17\right\rbrace

A∩B={12,16}A\cap B=\left\lbrace12,16\right\rbrace

Let U={10,11,12,13,14,15,16,17},A={10,12,14,16},B={11,12,15,16}U=\left\lbrace10,11,12,13,14,15,16,17\right\rbrace,A={\left\lbrace10,12,14,16\right\rbrace},B=\left\lbrace11,12,15,16\right\rbrace. Find: A′∩B=$$A^{\prime}$$\cap B=

A′∩B={11,15}$$A^{\prime}$$\cap B=\left\lbrace11,15\right\rbrace

A′∩B={12,16}$$A^{\prime}$$\cap B=\left\lbrace12,16\right\rbrace

A′∩B={13,17}$$A^{\prime}$$\cap B=\left\lbrace13,17\right\rbrace

A′∩B={10,14}$$A^{\prime}$$\cap B=\left\lbrace10,14\right\rbrace

Write the specified set using the roster method givenU={1,2,3,…,20},E={2,4,6,…,20},F={5,10,15,20}U=\left\lbrace{1,2,3,\ldots,20}\right\rbrace,E={\left\lbrace2,4,6,\ldots,20\right\rbrace},F={\left\lbrace5,10,15,20\right\rbrace}E∩FE∩F

E∩F={10,20}E\cap F=\left\lbrace10,20\right\rbrace

E∩F={2,4,6,8,10,12,14,16,18,20}E\cap F=\left\lbrace2,4,6,8,10,12,14,16,18,20\right\rbrace

E∩F={5,10,15,20}E\cap F=\left\lbrace5,10,15,20\right\rbrace

E∩F={2,5,10,15,20}E\cap F=\left\lbrace2,5,10,15,20\right\rbrace

Write the specified set using the roster method givenU={1,2,3,…,20},E={2,4,6,…,20},F={5,10,15,20}U=\left\lbrace{1,2,3,\ldots,20}\right\rbrace,E={\left\lbrace2,4,6,\ldots,20\right\rbrace},F={\left\lbrace5,10,15,20\right\rbrace}.F∩UF∩U

F∩U={5,10,15,20}F\cap U=\left\lbrace5,10,15,20\right\rbrace

F∩U={10,20}F\cap U=\left\lbrace10,20\right\rbrace

F∩U={1,2,3,…,20}F\cap U=\left\lbrace1,2,3,\ldots,20\right\rbrace

F∩U={2,4,6,8,10,12,14,16,18,20}F\cap U=\left\lbrace2,4,6,8,10,12,14,16,18,20\right\rbrace

Write the specified set using the roster method givenU={1,2,3,…,20},E={2,4,6,…,20},F={5,10,15,20}U=\left\lbrace{1,2,3,\ldots,20}\right\rbrace,E={\left\lbrace2,4,6,\ldots,20\right\rbrace},F={\left\lbrace5,10,15,20\right\rbrace}.E∩F′E∩F'

E∩F′={2,4,6,8,12,14,16,18}E\cap $$F^{\prime}$$=\left\lbrace2,4,6,8,12,14,16,18\right\rbrace

E∩F′={2,4,6,8,10,12,14,16,18,20}E\cap $$F^{\prime}$$=\left\lbrace2,4,6,8,10,12,14,16,18,20\right\rbrace

E∩F′={2,4,6,8,10,12,14,16}E\cap $$F^{\prime}$$=\left\lbrace2,4,6,8,10,12,14,16\right\rbrace

E∩F′={1,2,3,4,6,7,8,9,11,12,13,14,16,17,18,19}E\cap $$F^{\prime}$$=\left\lbrace1,2,3,4,6,7,8,9,11,12,13,14,16,17,18,19\right\rbrace

Write the specified set using the roster method givenU={1,2,3,…,20},E={2,4,6,…,20},F={5,10,15,20}U=\left\lbrace{1,2,3,\ldots,20}\right\rbrace,E={\left\lbrace2,4,6,\ldots,20\right\rbrace},F={\left\lbrace5,10,15,20\right\rbrace} .∅∩E∅∩E

∅∩E=∅\emptyset\cap E=\emptyset

∅∩E={2,4,6,…,20}\emptyset\cap E=\left\lbrace2,4,6,\ldots,20\right\rbrace

∅∩E={2,4,6,8,10,12,14,16,18,20}\emptyset\cap E=\left\lbrace2,4,6,8,10,12,14,16,18,20\right\rbrace

∅∩E={2,4,6,8,12,14,16,18}\emptyset\cap E=\left\lbrace2,4,6,8,12,14,16,18\right\rbrace

Label the Venn diagram with the elements that belong in each set using U={0,1,2,…,9},P={2,4,6,8},Q={3,6,9}U={\left\lbrace0,1,2,\ldots,9\right\rbrace},P={\left\lbrace2,4,6,8\right\rbrace},Q={\left\lbrace3,6,9\right\rbrace}. Then find & shade P∩QP∩Q.

P∩Q={6}P\cap Q=\left\lbrace6\right\rbrace

P∩Q={0,1,5,7}P\cap Q=\left\lbrace0,1,5,7\right\rbrace

Write the specified set using the roster method givenU={1,2,3,…,20},E={2,4,6,…,20},F={5,10,15,20}U={\left\lbrace1,2,3,\ldots,20\right\rbrace},E={\left\lbrace2,4,6,\ldots,20\right\rbrace},F={\left\lbrace5,10,15,20\right\rbrace} E∪FE∪F

E∪F={2,4,5,6,8,10,12,14,15,16,18,20}E\cup F=\left\lbrace2,4,5,6,8,10,12,14,15,16,18,20\right\rbrace

E∪F={2,4,6,8,12,14,16,18}E\cup F=\left\lbrace2,4,6,8,12,14,16,18\right\rbrace

E∪F={5,10,15,20}E\cup F=\left\lbrace5,10,15,20\right\rbrace

E∪F={1,2,3,…,20}E\cup F=\left\lbrace1,2,3,\ldots,20\right\rbrace

Write the specified set using the roster method givenU={1,2,3,…,20},E={2,4,6,…,20},F={5,10,15,20}U={\left\lbrace1,2,3,\ldots,20\right\rbrace},E={\left\lbrace2,4,6,\ldots,20\right\rbrace},F={\left\lbrace5,10,15,20\right\rbrace} E∪UE\cup U

E∪U={1,2,3,…,20}E\cup U=\left\lbrace1,2,3,\ldots,20\right\rbrace

E∪U={2,4,6,8,10,12,14,16,18,20}E\cup U=\left\lbrace2,4,6,8,10,12,14,16,18,20\right\rbrace

E∪U={2,4,5,6,8,10,12,14,15,16,18,20}E\cup U=\left\lbrace2,4,5,6,8,10,12,14,15,16,18,20\right\rbrace

E∪U={5,10,15,20}E\cup U=\left\lbrace5,10,15,20\right\rbrace

Write the specified set using the roster method givenU={1,2,3,…,20},E={2,4,6,…,20},F={5,10,15,20}U={\left\lbrace1,2,3,\ldots,20\right\rbrace},E={\left\lbrace2,4,6,\ldots,20\right\rbrace},F={\left\lbrace5,10,15,20\right\rbrace}E′∪$$FE^{\prime}$$\cup F

E′∪F={1,3,5,7,9,10,11,13,15,17,19,20}$$E^{\prime}$$\cup F=\left\lbrace1,3,5,7,9,10,11,13,15,17,19,20\right\rbrace

E′∪F={1,3,5,7,9,11,13,15,17,19}$$E^{\prime}$$\cup F=\left\lbrace1,3,5,7,9,11,13,15,17,19\right\rbrace

E′∪F={5,10,15,20}$$E^{\prime}$$\cup F=\left\lbrace5,10,15,20\right\rbrace

E′∪F={2,4,6,8,10,12,14,16,18,20}$$E^{\prime}$$\cup F=\left\lbrace2,4,6,8,10,12,14,16,18,20\right\rbrace

Write the specified set using the roster method givenU={1,2,3,…,20},E={2,4,6,…,20},F={5,10,15,20}U={\left\lbrace1,2,3,\ldots,20\right\rbrace},E={\left\lbrace2,4,6,\ldots,20\right\rbrace},F={\left\lbrace5,10,15,20\right\rbrace}.∅∪F∅∪F

∅∪F={5,10,15,20}\emptyset\cup F=\left\lbrace5,10,15,20\right\rbrace

∅∪F={1,2,3,…,20}\emptyset\cup F=\left\lbrace1,2,3,\ldots,20\right\rbrace

∅∪F={1,3,5,7,9,11,13,15,17,19}\emptyset\cup F=\left\lbrace1,3,5,7,9,11,13,15,17,19\right\rbrace

Label the Venn diagram with the elements that belong in each set using U={0,1,2,…,9},P={2,4,6,8},Q={3,6,9}U=\left\lbrace{0,1,2,\ldots,9}\right\rbrace,P={\left\lbrace2,4,6,8\right\rbrace},Q={\left\lbrace3,6,9\right\rbrace}. Then find & shade P∪Q.

P∪Q={2,4,8,6,3,9}P\cup Q=\left\lbrace2,4,8,6,3,9\right\rbrace

P∪Q={0,1,5,7}P\cup Q=\left\lbrace0,1,5,7\right\rbrace

P∪Q={6}P\cup Q=\left\lbrace6\right\rbrace

P∪Q={6,3,9}P\cup Q=\left\lbrace6,3,9\right\rbrace

Write the specified set using the roster method givenU={a,b,c,d,e,f,g,h},M={a,b,d},N={b,d,f,h}U={\left\lbrace a,b,c,d,e,f,g,h\right\rbrace},M={\left\lbrace a,b,d\right\rbrace},N=\left\lbrace{b,d,f,h}\right\rbrace.M∩(M∪N)M∩(M∪N)

M∩(M∪N)={a,b,d}M\cap(M\cup N)=\left\lbrace a,b,d\right\rbrace

M∩(M∪N)={b,d,f,h}M\cap(M\cup N)=\left\lbrace b,d,f,h\right\rbrace

M∩(M∪N)={a,b,d,f,h}M\cap(M\cup N)=\left\lbrace a,b,d,f,h\right\rbrace

M∩(M∪N)={a,b}M\cap(M\cup N)=\left\lbrace a,b\right\rbrace

Write the specified set using the roster method givenU={a,b,c,d,e,f,g,h},M={a,b,d},N={b,d,f,h}U={\left\lbrace a,b,c,d,e,f,g,h\right\rbrace},M={\left\lbrace a,b,d\right\rbrace},N=\left\lbrace{b,d,f,h}\right\rbrace.(M∪N′)∩(M′∪N)(M\cup $$N^{\prime}$$)\$$cap(M^{\prime}$$\cup N)

(M∪N′)∩(M′∪N)={b,c,d,e,g}(M\cup $$N^{\prime}$$)\$$cap(M^{\prime}$$\cup N)=\left\lbrace b,c,d,e,g\right\rbrace

(M∪N′)∩(M′∪N)={a,b,c,d,e,g}(M\cup $$N^{\prime}$$)\$$cap(M^{\prime}$$\cup N)=\left\lbrace a,b,c,d,e,g\right\rbrace

(M∪N′)∩(M′∪N)={a,b,d}(M\cup $$N^{\prime}$$)\$$cap(M^{\prime}$$\cup N)=\left\lbrace a,b,d\right\rbrace

(M∪N′)∩(M′∪N)={b,d,f,h}(M\cup $$N^{\prime}$$)\$$cap(M^{\prime}$$\cup N)=\left\lbrace b,d,f,h\right\rbrace

Write the specified set using the roster method given.U={a,b,c,d,e,f,g,h},M={a,b,d},N={b,d,f,h}U={\left\lbrace a,b,c,d,e,f,g,h\right\rbrace},M={\left\lbrace a,b,d\right\rbrace},N=\left\lbrace{b,d,f,h}\right\rbrace.N′∩(M∪$$U)N^{\prime}$$\cap(M\cup U)

N′∩(M∪U)={a,c,e,g}$$N^{\prime}$$\cap(M\cup U)=\left\lbrace a,c,e,g\right\rbrace

N′∩(M∪U)={b,d,f,h}$$N^{\prime}$$\cap(M\cup U)=\left\lbrace b,d,f,h\right\rbrace

N′∩(M∪U)={a,b,d}$$N^{\prime}$$\cap(M\cup U)=\left\lbrace a,b,d\right\rbrace

N′∩(M∪U)={a,b,c,d,e,f,g,h}$$N^{\prime}$$\cap(M\cup U)=\left\lbrace a,b,c,d,e,f,g,h\right\rbrace

Write the specified set using the roster method givenU={a,b,c,d,e,f,g,h},M={a,b,d},N={b,d,f,h}U={\left\lbrace a,b,c,d,e,f,g,h\right\rbrace},M={\left\lbrace a,b,d\right\rbrace},N=\left\lbrace{b,d,f,h}\right\rbrace.(∅∩M′)∪N(\emptyset\cap $$M^{\prime}$$)\cup N

(∅∩M′)∪N={b,d,f,h}(\emptyset\cap $$M^{\prime}$$)\cup N=\left\lbrace b,d,f,h\right\rbrace

(∅∩M′)∪N={a,b,d}(\emptyset\cap $$M^{\prime}$$)\cup N=\left\lbrace a,b,d\right\rbrace

(∅∩M′)∪N={∅}(\emptyset\cap $$M^{\prime}$$)\cup N=\left\lbrace\emptyset\right\rbrace

(∅∩M′)∪N={a,b,c,d,e,f,g,h}(\emptyset\cap $$M^{\prime}$$)\cup N=\left\lbrace a,b,c,d,e,f,g,h\right\rbrace

Use the Venn diagram to find each of the following.A′∩B′$$A^{\prime}$$\cap $$B^{\prime}$$

A′∩B′={0}$$A^{\prime}$$\cap $$B^{\prime}$$=\left\lbrace0\right\rbrace

A′∩B′={0,2,4}$$A^{\prime}$$\cap $$B^{\prime}$$=\left\lbrace0,2,4\right\rbrace

A′∩B′={2,4}$$A^{\prime}$$\cap $$B^{\prime}$$=\left\lbrace2,4\right\rbrace

A′∩B′={1,3,5,7,9}$$A^{\prime}$$\cap $$B^{\prime}$$=\left\lbrace1,3,5,7,9\right\rbrace

Use the Venn diagram to find each of the following.(A∪B)′(A\cup $$B)^{\prime}$$

(A∪B)′={0}(A\cup $$B)^{\prime}$$=\left\lbrace0\right\rbrace

(A∪B)′={0,2,4}(A\cup $$B)^{\prime}$$=\left\lbrace0,2,4\right\rbrace

(A∪B)′={2,4}(A\cup $$B)^{\prime}$$=\left\lbrace2,4\right\rbrace

(A∪B)′={1,3,5,7,9}(A\cup $$B)^{\prime}$$=\left\lbrace1,3,5,7,9\right\rbrace

Use the Venn diagram to find each of the following.(A∪B)′∪(A∩B)(A\cup $$B)^{\prime}$$\cup(A\cap B)

(A∪B)′∪(A∩B)={0,1,3,5,7,9}(A\cup $$B)^{\prime}$$\cup(A\cap B)=\left\lbrace0,1,3,5,7,9\right\rbrace

(A∪B)′∪(A∩B)={1,3,5,7,9}(A\cup $$B)^{\prime}$$\cup(A\cap B)=\left\lbrace1,3,5,7,9\right\rbrace

(A∪B)′∪(A∩B)={2,4,6,8}(A\cup $$B)^{\prime}$$\cup(A\cap B)=\left\lbrace2,4,6,8\right\rbrace

(A∪B)′∪(A∩B)={2,4,1,3,5,7,9,6,8}(A\cup $$B)^{\prime}$$\cup(A\cap B)=\left\lbrace2,4,1,3,5,7,9,6,8\right\rbrace

2. Sets

Intersections and Unions

2. Sets

Intersections and Unions: Videos & Practice Problems

Video Lessons

Concept

Intersections

Video duration:

Problem

Let $U = {10, 11, 12, 13, 14, 15, 16, 17}, A = {10, 12, 14, 16}, B = {11, 12, 15, 16}$ . Find:
$A \cap B =$
$Venn diagram showing two sets A and B within universal set U, illustrating their intersection.$

$Venn diagram showing universal set U with elements 10-17, sets A and B, and their intersection containing 12 and 16.$

$A \cap B = {13, 17}$

$Venn diagram with sets A and B inside universal set U, highlighting intersection area containing 13 and 17.$

$A \cap B = {13, 17}$

$Venn diagram showing sets A and B within universal set U, highlighting intersection with elements 12 and 16.$

$A \cap B = {12, 16}$

$Venn diagram with two overlapping circles labeled A and B inside a yellow rectangle labeled U, showing numbers in each section.$

$A \cap B = {12, 16}$

Problem

Let $U = {10, 11, 12, 13, 14, 15, 16, 17}, A = {10, 12, 14, 16}, B = {11, 12, 15, 16}$ . Find:
$A^{'} \cap B =$
$Venn diagram showing universal set U with subsets A and B overlapping, illustrating set intersections and unions.$

$Venn diagram showing sets A and B within universal set U, highlighting the intersection and elements outside A but in B.$

$A^{'} \cap B = {11, 15}$

$Venn diagram with sets A and B inside universal set U, showing elements and their intersections.$

$A^{'} \cap B = {12, 16}$

$Venn diagram showing sets A and B within universal set U, with elements labeled and overlapping values 12 and 16.$

$A^{'} \cap B = {13, 17}$

$Venn diagram showing universal set U with subsets A (green) and B (blue), highlighting elements 12 and 16 in their intersection.$

$A^{'} \cap B = {10, 14}$

Problem

Write the specified set using the roster method given
$U = {1, 2, 3, \dots, 20}, E = {2, 4, 6, \dots, 20}, F = {5, 10, 15, 20}$
$E \cap F$

$E \cap F = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20}$

$E \cap F = {5, 10, 15, 20}$

$E \cap F = {2, 5, 10, 15, 20}$

$E \cap F = {10, 20}$

Problem

Write the specified set using the roster method given
$U = {1, 2, 3, \dots, 20}, E = {2, 4, 6, \dots, 20}, F = {5, 10, 15, 20}$ .
$F \cap U$

$F \cap U = {10, 20}$

$F \cap U = {5, 10, 15, 20}$

$F \cap U = {1, 2, 3, \dots, 20}$

$F \cap U = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20}$

Problem

Write the specified set using the roster method given
$U = {1, 2, 3, \dots, 20}, E = {2, 4, 6, \dots, 20}, F = {5, 10, 15, 20}$ .
$E \cap F^{'}$

$E \cap F^{'} = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20}$

$E \cap F^{'} = {2, 4, 6, 8, 10, 12, 14, 16}$

$E \cap F^{'} = {2, 4, 6, 8, 12, 14, 16, 18}$

$E \cap F^{'} = {1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19}$

Problem

Write the specified set using the roster method given
$U = {1, 2, 3, \dots, 20}, E = {2, 4, 6, \dots, 20}, F = {5, 10, 15, 20}$ .
$∅ \cap E$

$\emptyset \cap E = {2, 4, 6, \dots, 20}$

$\emptyset \cap E = \emptyset$

$\emptyset \cap E = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20}$

$\emptyset \cap E = {2, 4, 6, 8, 12, 14, 16, 18}$

Example

Intersections Example 1

Video duration:

Problem

Label the Venn diagram with the elements that belong in each set using $U = {0, 1, 2, \dots, 9}, P = {2, 4, 6, 8}, Q = {3, 6, 9}$ . Then find & shade $P \cap Q$ .
$Venn diagram with sets U={0-9}, P={2,4,6,8}, Q={3,6,9}, showing intersection P∩Q shaded at element 6.$

$Venn diagram with sets P and Q labeled with elements from 0 to 9, highlighting the intersection containing 6.$

$P \cap Q = {6}$

$Venn diagram with sets P={2,4,6,8} and Q={3,6,9} labeled, shading the intersection containing 6.$

$P \cap Q = {6}$

$Venn diagram with sets P and Q labeled with elements from 0 to 9, highlighting the intersection containing 6.$

$P \cap Q = {0, 1, 5, 7}$

$Venn diagram with sets P={2,4,6,8} and Q={3,6,9} labeled, shading the intersection containing 6.$

$P \cap Q = {0, 1, 5, 7}$

Problem

Shade each of the following:
$M \cap N$
$Venn diagram with two overlapping circles labeled M and N inside a rectangle labeled U, showing the intersection area.$

$Venn diagram with two overlapping circles labeled M and N inside a yellow rectangle labeled U, showing their intersection.$

$Two overlapping circles labeled M and N inside a rectangle labeled U, with both circles fully shaded in yellow.$

$Venn diagram with two overlapping yellow circles labeled M and N inside a rectangle labeled U.$

$Venn diagram with two overlapping circles labeled M and N, showing the intersection area shaded in yellow.$

Problem

Shade each of the following:
$M^{'} \cap N$
$Venn diagram with two overlapping circles labeled M and N inside a rectangle labeled U, illustrating set intersections.$

$Venn diagram with two overlapping circles labeled M and N, highlighting the intersection area in yellow.$

$Venn diagram with two circles labeled M and N inside a rectangle U, with the M circle shaded yellow except the overlap.$

$Venn diagram with two overlapping circles labeled M and N, shading the area of circle N only.$

$Venn diagram with two overlapping circles labeled M and N inside a yellow rectangle labeled U.$

Problem

Shade each of the following:
$M \cap N^{'}$
$Venn diagram with two overlapping circles labeled M and N inside a rectangle labeled U, highlighting M intersecting N complement.$

$Venn diagram with two overlapping circles labeled M and N, highlighting the intersection area in yellow.$

$Venn diagram with two circles labeled M and N inside a rectangle U, with the M circle shaded yellow except the overlap.$

$Venn diagram with two overlapping circles labeled M and N, shading the area of circle N only.$

$Venn diagram with two overlapping circles labeled M and N inside a yellow rectangle labeled U.$

Problem

Shade each of the following:
$M^{'} \cap N^{'}$
$Venn diagram with two overlapping circles labeled M and N inside a rectangle labeled U, showing sets and their complements.$

$Venn diagram with two overlapping circles labeled M and N, highlighting the intersection area in yellow.$

$Venn diagram with two circles labeled M and N inside a rectangle U, with the M circle shaded yellow except the overlap.$

$Venn diagram with two overlapping circles labeled M and N, shading the area of circle N only.$

$Venn diagram with two overlapping circles labeled M and N inside a yellow rectangle labeled U.$

Concept

Unions

Video duration:

Problem

Write the specified set using the roster method given
$U = {1, 2, 3, \dots, 20}, E = {2, 4, 6, \dots, 20}, F = {5, 10, 15, 20}$
$E \cup F$

$E \cup F = {2, 4, 6, 8, 12, 14, 16, 18}$

$E \cup F = {5, 10, 15, 20}$

$E \cup F = {1, 2, 3, \dots, 20}$

$E \cup F = {2, 4, 5, 6, 8, 10, 12, 14, 15, 16, 18, 20}$

Problem

Write the specified set using the roster method given
$U = {1, 2, 3, \dots, 20}, E = {2, 4, 6, \dots, 20}, F = {5, 10, 15, 20}$
$E \cup U$

$E \cup U = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20}$

$E \cup U = {1, 2, 3, \dots, 20}$

$E \cup U = {2, 4, 5, 6, 8, 10, 12, 14, 15, 16, 18, 20}$

$E \cup U = {5, 10, 15, 20}$

Problem

Write the specified set using the roster method given
$U = {1, 2, 3, \dots, 20}, E = {2, 4, 6, \dots, 20}, F = {5, 10, 15, 20}$
$E^{'} \cup F$

$E^{'} \cup F = {1, 3, 5, 7, 9, 10, 11, 13, 15, 17, 19, 20}$

$E^{'} \cup F = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19}$

$E^{'} \cup F = {5, 10, 15, 20}$

$E^{'} \cup F = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20}$

Problem

Write the specified set using the roster method given
$U = {1, 2, 3, \dots, 20}, E = {2, 4, 6, \dots, 20}, F = {5, 10, 15, 20}$ .
$∅ \cup F$

$\emptyset \cup F = \emptyset$

$\emptyset \cup F = {1, 2, 3, \dots, 20}$

$\emptyset \cup F = {5, 10, 15, 20}$

$\emptyset \cup F = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19}$

Example

Unions Example 2

Video duration:

Problem

Label the Venn diagram with the elements that belong in each set using $U = {0, 1, 2, \dots, 9}, P = {2, 4, 6, 8}, Q = {3, 6, 9}$ . Then find & shade P∪Q.
$Venn diagram with universal set 0-9, set P with 2,4,6,8, set Q with 3,6,9, showing their union shaded.$

$Venn diagram with sets P={2,4,6,8} and Q={3,6,9} labeled and union P∪Q shaded within universal set U={0–9}.$

$P \cup Q = {0, 1, 5, 7}$

$Venn diagram with sets P and Q labeled, elements placed, and the union P∪Q shaded in yellow.$

$P \cup Q = {6}$

$Venn diagram with universal set 0-9, sets P={2,4,6,8} and Q={3,6,9}, shaded union of P and Q.$

$P \cup Q = {2, 4, 8, 6, 3, 9}$

$Venn diagram with sets P and Q labeled, elements placed, and union P∪Q shaded in yellow.$

$P \cup Q = {6, 3, 9}$

Problem

Shade each of the following:
$M \cup N$
$Venn diagram with two overlapping circles labeled M and N inside a rectangle labeled U, showing the union of M and N shaded.$

$Venn diagram with two circles labeled M and N inside a rectangle U, shading the area covering M and the intersection with N.$

$Venn diagram with two overlapping circles labeled M and N, shading all areas except their intersection.$

$Venn diagram with two circles labeled M and N inside a rectangle labeled U, with the area outside both circles shaded yellow.$

$Two overlapping yellow circles labeled M and N inside a rectangle labeled U, representing their union shaded.$

Problem

Shade each of the following:
$M^{'} \cup N$
$Venn diagram with two overlapping circles labeled M and N inside a rectangle labeled U, illustrating set operations.$

$Venn diagram with two circles labeled M and N inside a rectangle U, shading the area covering M and the intersection with N.$

$Venn diagram with two overlapping circles labeled M and N, shading all areas except their intersection.$

$Venn diagram with two circles labeled M and N inside a rectangle labeled U, with the area outside both circles shaded yellow.$

$Two overlapping yellow circles labeled M and N inside a rectangle labeled U, representing their union shaded.$

Problem

Shade each of the following:
$M \cup N^{'}$
$Venn diagram with two overlapping circles labeled M and N inside a rectangle labeled U.$

$Venn diagram with two circles labeled M and N inside a rectangle U, shading the area covering M and the intersection with N.$

$Venn diagram with two overlapping circles labeled M and N, shading all areas except their intersection.$

$Venn diagram with two circles labeled M and N inside a rectangle labeled U, with the area outside both circles shaded yellow.$

$Two overlapping yellow circles labeled M and N inside a rectangle labeled U, representing their union shaded.$

Problem

Shade each of the following:
$M^{'} \cup N^{'}$
$Venn diagram with two overlapping circles labeled M and N inside a rectangle labeled U, highlighting the union of their complements.$

$Venn diagram with two circles labeled M and N inside a rectangle U, shading the area covering M and the intersection with N.$

$Venn diagram with two overlapping circles labeled M and N, shading all areas except their intersection.$

$Venn diagram with two circles labeled M and N inside a rectangle labeled U, with the area outside both circles shaded yellow.$

$Two overlapping yellow circles labeled M and N inside a rectangle labeled U, representing their union shaded.$

Example

Unions Example 3

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Concept

Multiple Set Operations

Video duration:

Problem

Shade each of the following on the Venn diagrams.
$P \cap (P \cap Q^{'})$
$Venn diagram with two overlapping circles labeled P and Q inside a rectangle labeled U, showing set intersections to shade.$

$Venn diagram with two circles labeled P and Q, highlighting the area of circle Q shaded in yellow.$

$Venn diagram with two sets, P shaded yellow and Q unshaded, illustrating the intersection of P and complement of Q.$

$Venn diagram with two circles labeled P and Q, shading the area representing P intersecting P intersecting Q complement.$

$Venn diagram with two circles labeled P and Q, Q circle shaded, representing set union and intersection concepts.$

Problem

Shade each of the following on the Venn diagrams.
$(P^{'} \cap Q) \cup (P \cap Q)$
$Venn diagram with two overlapping circles labeled P and Q inside a rectangle labeled U, showing shaded regions for set expression.$

$Venn diagram with two circles labeled P and Q, highlighting the area of circle Q shaded in yellow.$

$Venn diagram with two sets, P shaded yellow and Q unshaded, illustrating the intersection of P and complement of Q.$

$Venn diagram with two circles labeled P and Q, shading the area representing P intersecting P intersecting Q complement.$

$Venn diagram with two circles labeled P and Q, Q circle shaded, representing set union and intersection concepts.$

Example

Multiple Set Operations Example 4

Video duration:

Problem

Write the specified set using the roster method given
$U = {a, b, c, d, e, f, g, h}, M = {a, b, d}, N = {b, d, f, h}$ .
$M \cap (M \cup N)$

$M \cap (M \cup N) = {b, d, f, h}$

$M \cap (M \cup N) = {a, b, d}$

$M \cap (M \cup N) = {a, b, d, f, h}$

$M \cap (M \cup N) = {a, b}$

Problem

Write the specified set using the roster method given
$U = {a, b, c, d, e, f, g, h}, M = {a, b, d}, N = {b, d, f, h}$ .
$(M \cup N^{'}) \cap (M^{'} \cup N)$

$(M \cup N^{'}) \cap (M^{'} \cup N) = {a, b, c, d, e, g}$

$(M \cup N^{'}) \cap (M^{'} \cup N) = {a, b, d}$

$(M \cup N^{'}) \cap (M^{'} \cup N) = {b, c, d, e, g}$

$(M \cup N^{'}) \cap (M^{'} \cup N) = {b, d, f, h}$

Problem

Write the specified set using the roster method given.
$U = {a, b, c, d, e, f, g, h}, M = {a, b, d}, N = {b, d, f, h}$ .
$N^{'} \cap (M \cup U)$

$N^{'} \cap (M \cup U) = {a, c, e, g}$

$N^{'} \cap (M \cup U) = {b, d, f, h}$

$N^{'} \cap (M \cup U) = {a, b, d}$

$N^{'} \cap (M \cup U) = {a, b, c, d, e, f, g, h}$

Problem

Write the specified set using the roster method given
$U = {a, b, c, d, e, f, g, h}, M = {a, b, d}, N = {b, d, f, h}$ .
$(\emptyset \cap M^{'}) \cup N$

$(\emptyset \cap M^{'}) \cup N = {a, b, d}$

$(\emptyset \cap M^{'}) \cup N = {\emptyset}$

$(\emptyset \cap M^{'}) \cup N = {a, b, c, d, e, f, g, h}$

$(\emptyset \cap M^{'}) \cup N = {b, d, f, h}$

Problem

Use the Venn diagram to find each of the following.
$A^{'} \cap B^{'}$
$Venn diagram with two overlapping circles labeled A and B containing numbers representing elements in each set and their intersections.$

$A^{'} \cap B^{'} = {0, 2, 4}$

$A^{'} \cap B^{'} = {2, 4}$

$A^{'} \cap B^{'} = {1, 3, 5, 7, 9}$

$A^{'} \cap B^{'} = {0}$

Problem

Use the Venn diagram to find each of the following.
$(A \cup B)^{'}$
$Venn diagram with two overlapping circles labeled A and B containing numbers representing elements in each set and their intersection.$

$(A \cup B)^{'} = {0, 2, 4}$

$(A \cup B)^{'} = {2, 4}$

$(A \cup B)^{'} = {1, 3, 5, 7, 9}$

$(A \cup B)^{'} = {0}$

Problem

Use the Venn diagram to find each of the following.
$(A \cup B)^{'} \cup (A \cap B)$
$Venn diagram with two overlapping circles labeled A and B containing numbers, illustrating set operations.$

$(A \cup B)^{'} \cup (A \cap B) = {1, 3, 5, 7, 9}$

$(A \cup B)^{'} \cup (A \cap B) = {2, 4, 6, 8}$

$(A \cup B)^{'} \cup (A \cap B) = {0, 1, 3, 5, 7, 9}$

$(A \cup B)^{'} \cup (A \cap B) = {2, 4, 1, 3, 5, 7, 9, 6, 8}$

Your Liberal Arts Math tutors

Patrick Ford

Physics and Math Lead Instructor

Intersections and Unions: Videos & Practice Problems

Intersections

Let U={10,11,12,13,14,15,16,17},A={10,12,14,16},B={11,12,15,16}U=\(\left\[\lbrace\)10,11,12,13,14,15,16,17\(\right\]\rbrace\),A={\(\left\[\lbrace\)10,12,14,16\(\right\]\rbrace\)},B=\(\left\[\lbrace\)11,12,15,16\(\right\]\rbrace\). Find:A∩B=A∩B=

Let U={10,11,12,13,14,15,16,17},A={10,12,14,16},B={11,12,15,16}U=\(\left\[\lbrace\)10,11,12,13,14,15,16,17\(\right\]\rbrace\),A={\(\left\[\lbrace\)10,12,14,16\(\right\]\rbrace\)},B=\(\left\[\lbrace\)11,12,15,16\(\right\]\rbrace\). Find: A′∩B=A^{\(\prime\)}\(\cap\) B=

Write the specified set using the roster method givenU={1,2,3,…,20},E={2,4,6,…,20},F={5,10,15,20}U=\(\left\]\lbrace{1,2,3,\ldots,20}\[\right\]\rbrace\),E={\(\left\[\lbrace\)2,4,6,\(\ldots\),20\(\right\]\rbrace\)},F={\(\left\[\lbrace\)5,10,15,20\(\right\]\rbrace\)}E∩FE∩F

Write the specified set using the roster method givenU={1,2,3,…,20},E={2,4,6,…,20},F={5,10,15,20}U=\(\left\]\lbrace{1,2,3,\ldots,20}\[\right\]\rbrace\),E={\(\left\[\lbrace\)2,4,6,\(\ldots\),20\(\right\]\rbrace\)},F={\(\left\[\lbrace\)5,10,15,20\(\right\]\rbrace\)}.F∩UF∩U

Write the specified set using the roster method givenU={1,2,3,…,20},E={2,4,6,…,20},F={5,10,15,20}U=\(\left\]\lbrace{1,2,3,\ldots,20}\[\right\]\rbrace\),E={\(\left\[\lbrace\)2,4,6,\(\ldots\),20\(\right\]\rbrace\)},F={\(\left\[\lbrace\)5,10,15,20\(\right\]\rbrace\)}.E∩F′E∩F'

Write the specified set using the roster method givenU={1,2,3,…,20},E={2,4,6,…,20},F={5,10,15,20}U=\(\left\]\lbrace{1,2,3,\ldots,20}\[\right\]\rbrace\),E={\(\left\[\lbrace\)2,4,6,\(\ldots\),20\(\right\]\rbrace\)},F={\(\left\[\lbrace\)5,10,15,20\(\right\]\rbrace\)} .∅∩E∅∩E

Intersections Example 1

Label the Venn diagram with the elements that belong in each set using U={0,1,2,…,9},P={2,4,6,8},Q={3,6,9}U={\(\left\[\lbrace\)0,1,2,\(\ldots\),9\(\right\]\rbrace\)},P={\(\left\[\lbrace\)2,4,6,8\(\right\]\rbrace\)},Q={\(\left\[\lbrace\)3,6,9\(\right\]\rbrace\)}. Then find & shade P∩QP∩Q.

Shade each of the following:M∩NM∩N

Shade each of the following:M′∩NM'∩N

Shade each of the following:M∩N′M∩N'

Shade each of the following:M′∩N′M'∩N'

Unions

Write the specified set using the roster method givenU={1,2,3,…,20},E={2,4,6,…,20},F={5,10,15,20}U={\(\left\[\lbrace\)1,2,3,\(\ldots\),20\(\right\]\rbrace\)},E={\(\left\[\lbrace\)2,4,6,\(\ldots\),20\(\right\]\rbrace\)},F={\(\left\[\lbrace\)5,10,15,20\(\right\]\rbrace\)} E∪FE∪F

Write the specified set using the roster method givenU={1,2,3,…,20},E={2,4,6,…,20},F={5,10,15,20}U={\(\left\[\lbrace\)1,2,3,\(\ldots\),20\(\right\]\rbrace\)},E={\(\left\[\lbrace\)2,4,6,\(\ldots\),20\(\right\]\rbrace\)},F={\(\left\[\lbrace\)5,10,15,20\(\right\]\rbrace\)} E∪UE\(\cup\) U

Write the specified set using the roster method givenU={1,2,3,…,20},E={2,4,6,…,20},F={5,10,15,20}U={\(\left\[\lbrace\)1,2,3,\(\ldots\),20\(\right\]\rbrace\)},E={\(\left\[\lbrace\)2,4,6,\(\ldots\),20\(\right\]\rbrace\)},F={\(\left\[\lbrace\)5,10,15,20\(\right\]\rbrace\)}.∅∪F∅∪F

Unions Example 2

Label the Venn diagram with the elements that belong in each set using U={0,1,2,…,9},P={2,4,6,8},Q={3,6,9}U=\(\left\]\lbrace{0,1,2,\ldots,9}\[\right\]\rbrace\),P={\(\left\[\lbrace\)2,4,6,8\(\right\]\rbrace\)},Q={\(\left\[\lbrace\)3,6,9\(\right\]\rbrace\)}. Then find & shade P∪Q.

Shade each of the following:M∪NM\(\cup\) N

Shade each of the following:M′∪NM^{\(\prime\)}\(\cup\) N

Shade each of the following:M∪N′M\(\cup\) N^{\(\prime\)}

Shade each of the following:M′∪N′M^{\(\prime\)}\(\cup\) N^{\(\prime\)}

Unions Example 3

Multiple Set Operations

Shade each of the following on the Venn diagrams.P∩(P∩Q′)P∩(P∩Q')

Shade each of the following on the Venn diagrams.(P′∩Q)∪(P∩Q)(P^{\(\prime\)}\(\cap\) Q)\(\cup\)(P\(\cap\) Q)

Multiple Set Operations Example 4

Write the specified set using the roster method givenU={a,b,c,d,e,f,g,h},M={a,b,d},N={b,d,f,h}U={\(\left\]\lbrace\) a,b,c,d,e,f,g,h\(\right\[\rbrace\)},M={\(\left\]\lbrace\) a,b,d\(\right\[\rbrace\)},N=\(\left\]\lbrace{b,d,f,h}\[\right\]\rbrace\).M∩(M∪N)M∩(M∪N)

Write the specified set using the roster method given.U={a,b,c,d,e,f,g,h},M={a,b,d},N={b,d,f,h}U={\(\left\]\lbrace\) a,b,c,d,e,f,g,h\(\right\[\rbrace\)},M={\(\left\]\lbrace\) a,b,d\(\right\[\rbrace\)},N=\(\left\]\lbrace{b,d,f,h}\[\right\]\rbrace\).N′∩(M∪U)N^{\(\prime\)}\(\cap\)(M\(\cup\) U)

Use the Venn diagram to find each of the following.A′∩B′A^{\(\prime\)}\(\cap\) B^{\(\prime\)}

Use the Venn diagram to find each of the following.(A∪B)′(A\(\cup\) B)^{\(\prime\)}

Use the Venn diagram to find each of the following.(A∪B)′∪(A∩B)(A\(\cup\) B)^{\(\prime\)}\(\cup\)(A\(\cap\) B)

Your Liberal Arts Math tutors

Let $U = {10, 11, 12, 13, 14, 15, 16, 17}, A = {10, 12, 14, 16}, B = {11, 12, 15, 16}$ . Find:
$A \cap B =$
$Venn diagram showing two sets A and B within universal set U, illustrating their intersection.$

Let $U = {10, 11, 12, 13, 14, 15, 16, 17}, A = {10, 12, 14, 16}, B = {11, 12, 15, 16}$ . Find:
$A^{'} \cap B =$
$Venn diagram showing universal set U with subsets A and B overlapping, illustrating set intersections and unions.$

Write the specified set using the roster method given
$U = {1, 2, 3, \dots, 20}, E = {2, 4, 6, \dots, 20}, F = {5, 10, 15, 20}$
$E \cap F$

Write the specified set using the roster method given
$U = {1, 2, 3, \dots, 20}, E = {2, 4, 6, \dots, 20}, F = {5, 10, 15, 20}$ .
$F \cap U$

Write the specified set using the roster method given
$U = {1, 2, 3, \dots, 20}, E = {2, 4, 6, \dots, 20}, F = {5, 10, 15, 20}$ .
$E \cap F^{'}$

Write the specified set using the roster method given
$U = {1, 2, 3, \dots, 20}, E = {2, 4, 6, \dots, 20}, F = {5, 10, 15, 20}$ .
$∅ \cap E$

Label the Venn diagram with the elements that belong in each set using $U = {0, 1, 2, \dots, 9}, P = {2, 4, 6, 8}, Q = {3, 6, 9}$ . Then find & shade $P \cap Q$ .
$Venn diagram with sets U={0-9}, P={2,4,6,8}, Q={3,6,9}, showing intersection P∩Q shaded at element 6.$

Shade each of the following:
$M \cap N$
$Venn diagram with two overlapping circles labeled M and N inside a rectangle labeled U, showing the intersection area.$

Shade each of the following:
$M^{'} \cap N$
$Venn diagram with two overlapping circles labeled M and N inside a rectangle labeled U, illustrating set intersections.$

Shade each of the following:
$M \cap N^{'}$
$Venn diagram with two overlapping circles labeled M and N inside a rectangle labeled U, highlighting M intersecting N complement.$

Shade each of the following:
$M^{'} \cap N^{'}$
$Venn diagram with two overlapping circles labeled M and N inside a rectangle labeled U, showing sets and their complements.$

Write the specified set using the roster method given
$U = {1, 2, 3, \dots, 20}, E = {2, 4, 6, \dots, 20}, F = {5, 10, 15, 20}$
$E \cup F$

Write the specified set using the roster method given
$U = {1, 2, 3, \dots, 20}, E = {2, 4, 6, \dots, 20}, F = {5, 10, 15, 20}$
$E \cup U$

Write the specified set using the roster method given
$U = {1, 2, 3, \dots, 20}, E = {2, 4, 6, \dots, 20}, F = {5, 10, 15, 20}$ .
$∅ \cup F$

Label the Venn diagram with the elements that belong in each set using $U = {0, 1, 2, \dots, 9}, P = {2, 4, 6, 8}, Q = {3, 6, 9}$ . Then find & shade P∪Q.
$Venn diagram with universal set 0-9, set P with 2,4,6,8, set Q with 3,6,9, showing their union shaded.$

Shade each of the following:
$M \cup N$
$Venn diagram with two overlapping circles labeled M and N inside a rectangle labeled U, showing the union of M and N shaded.$

Shade each of the following:
$M^{'} \cup N$
$Venn diagram with two overlapping circles labeled M and N inside a rectangle labeled U, illustrating set operations.$

Shade each of the following:
$M \cup N^{'}$
$Venn diagram with two overlapping circles labeled M and N inside a rectangle labeled U.$

Shade each of the following:
$M^{'} \cup N^{'}$
$Venn diagram with two overlapping circles labeled M and N inside a rectangle labeled U, highlighting the union of their complements.$

Shade each of the following on the Venn diagrams.
$P \cap (P \cap Q^{'})$
$Venn diagram with two overlapping circles labeled P and Q inside a rectangle labeled U, showing set intersections to shade.$

Shade each of the following on the Venn diagrams.
$(P^{'} \cap Q) \cup (P \cap Q)$
$Venn diagram with two overlapping circles labeled P and Q inside a rectangle labeled U, showing shaded regions for set expression.$

Write the specified set using the roster method given
$U = {a, b, c, d, e, f, g, h}, M = {a, b, d}, N = {b, d, f, h}$ .
$M \cap (M \cup N)$

Write the specified set using the roster method given.
$U = {a, b, c, d, e, f, g, h}, M = {a, b, d}, N = {b, d, f, h}$ .
$N^{'} \cap (M \cup U)$

Use the Venn diagram to find each of the following.
$A^{'} \cap B^{'}$
$Venn diagram with two overlapping circles labeled A and B containing numbers representing elements in each set and their intersections.$

Use the Venn diagram to find each of the following.
$(A \cup B)^{'}$
$Venn diagram with two overlapping circles labeled A and B containing numbers representing elements in each set and their intersection.$

Use the Venn diagram to find each of the following.
$(A \cup B)^{'} \cup (A \cap B)$
$Venn diagram with two overlapping circles labeled A and B containing numbers, illustrating set operations.$