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1. If [tex]$U = \{2,4,6,8,10,12,14\}[tex]$[/tex], [tex]A = \{2,6,10,12\}[/tex], and [tex]B = \{6,10,12,14\}$[/tex][/tex], prove that:

[tex]\rightarrow (A \cup B)[/tex]

[tex]\begin{array}{l}
A = \{2,6,10,12\} \\
B = \{6,10,12,14\} \\
\text{so } A \cup B = \{2,6,10,12,14\} \\
\end{array}[/tex]

[tex]= A \cap B[/tex]



Answer :

GinnyAnswer
To solve the given problem, let's go through the required steps to prove the union and intersection of the sets \( A \) and \( B \).

Given:
- \( U = \{2, 4, 6, 8, 10, 12, 14\} \)
- \( A = \{2, 6, 10, 12\} \)
- \( B = \{6, 10, 12, 14\} \)

### Finding the Union \((A \cup B)\):

The union of two sets \(A\) and \(B\) is the set of elements that are in either \( A \) or \( B \) or in both.

To compute \( A \cup B \), we combine all unique elements from both sets \(A\) and \(B\):

[tex]\[ A \cup B = \{2, 6, 10, 12\} \cup \{6, 10, 12, 14\} \][/tex]

Combine all the elements:

[tex]\[ A \cup B = \{2, 6, 10, 12, 14\} \][/tex]

### Finding the Intersection \((A \cap B)\):

The intersection of two sets \(A\) and \(B\) is the set of elements that are in both \(A\) and \(B\).

To compute \( A \cap B \), we identify all elements that are common to both sets \(A\) and \(B\):

[tex]\[ A \cap B = \{2, 6, 10, 12\} \cap \{6, 10, 12, 14\} \][/tex]

Identify the common elements:

[tex]\[ A \cap B = \{6, 10, 12\} \][/tex]

### Summary

Thus, the union and intersection of sets \( A \) and \( B \) are:

[tex]\[ A \cup B = \{2, 6, 10, 12, 14\} \][/tex]
[tex]\[ A \cap B = \{6, 10, 12\} \][/tex]

This concludes our solution.

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