Question

use the following venn diagram to find ((a cap b)^c).
venn diagram: universal set (rectangle) with circle ( a ) (purple) containing ( a, b, c ); intersection (blue - green) containing ( d, e ); circle ( b ) (green) containing ( f, g, h, i ); outside both circles (in rectangle) containing ( j, k, l, m ).
multiple - choice options: ({a, b, c, f, g, h, i, j, k, l, m}), ({a, e, f, i, j, k, p, q, r, x, y}), ({b, c, e, g, h, l, p, r, s, w, z}), ({b, c, d, e, f, q, r, s, t, y, z}), ({b, f, i, k, l, m, p, q, r, y, z}), (emptyset)

Explanation:

Step1: Find \( A \cap B \)

The intersection \( A \cap B \) is the set of elements common to both \( A \) and \( B \). From the Venn diagram, the overlapping region (intersection) of \( A \) and \( B \) contains elements \( d, e \). So \( A \cap B=\{d, e\} \).

Step2: Find the complement \( (A \cap B)^c \)

The complement of a set \( S \) (denoted \( S^c \)) is the set of all elements in the universal set \( U \) that are not in \( S \). First, identify the universal set \( U \) from the Venn diagram. The universal set includes all elements in the rectangle: \( A \)'s non - overlapping part (\( a, b, c \)), \( B \)'s non - overlapping part (\( f, g, h, i \)), the intersection (\( d, e \)), and the elements outside both circles (\( j, k, l, m \)). So \( U=\{a, b, c, d, e, f, g, h, i, j, k, l, m\} \).

To find \( (A \cap B)^c \), we remove the elements of \( A \cap B \) (i.e., \( d, e \)) from \( U \). So \( (A \cap B)^c=\{a, b, c, f, g, h, i, j, k, l, m\} \).

Answer:

\(\{a, b, c, f, g, h, i, j, k, l, m\}\) (the first option)