Determine whether each statement is true or false. [6, 12, 14, 16} ∪ {6, 14, 19} = {6, 14}

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Recall that the union of two sets $ A $ and $ B $, denoted by $ A \cup B $, is the set containing all elements that are in $ A $, in $ B $, or in both.

Identify the two sets given: $ A = \{6, 12, 14, 16\} $ and $ B = \{6, 14, 19\} $.

List all unique elements from both sets combined: start with all elements of $ A $, then add elements from $ B $ that are not already in $ A $.

Combine the elements: $ \{6, 12, 14, 16\} \cup \{6, 14, 19\} = \{6, 12, 14, 16, 19\} $.

Compare the resulting union $ \{6, 12, 14, 16, 19\} $ with the set given on the right side of the equation $ \{6, 14\} $ to determine if the statement is true or false.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Set Union

The union of two sets combines all unique elements from both sets into one set. For example, the union of {1, 2} and {2, 3} is {1, 2, 3}. It includes every element that appears in either set without duplication.

Set Notation and Elements

Sets are collections of distinct elements, usually enclosed in curly braces. Understanding how to read and interpret set notation is essential, including recognizing elements and how they are grouped or listed.

Equality of Sets

Two sets are equal if and only if they contain exactly the same elements. Order and repetition do not matter. For example, {1, 2, 3} equals {3, 2, 1}, but not {1, 2}.

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Factor each polynomial. See Examples 5 and 6. $9 m^{2} - n^{2} - 2 n - 1$