Question

use the following venn diagram to find ( a^c cap b^c ).
venn diagram: circle a (purple) contains a, b, c; intersection with circle b (green) contains d, e; circle b contains f, g, h, i; region outside both circles contains j, k, l, m.
options:
○ {c, m, x}
○ {j, k, l, m}
○ {c, e, o}
○ {d, i, r}
○ {d, j, p}
○ ∅

Explanation:

Step1: Recall De Morgan's Law

By De Morgan's Law, \( A^c \cap B^c=(A \cup B)^c \), which means the elements not in \( A \) and not in \( B \) (the complement of the union of \( A \) and \( B \)).

Step2: Identify elements in \( A \), \( B \), and the universal set

• Set \( A \) (purple circle) contains: \( \{a, b, c, d, e\} \)
• Set \( B \) (green circle) contains: \( \{d, e, f, g, h, i\} \)
• The universal set (the rectangle) contains: \( \{a, b, c, d, e, f, g, h, i, j, k, l, m\} \) (assuming no other elements like \( x, o, r, p \) as they are not in the Venn diagram shown)

Step3: Find \( (A \cup B)^c \)

First, find \( A \cup B \): \( A \cup B=\{a, b, c, d, e, f, g, h, i\} \) (all elements in \( A \) or \( B \)). Then, the complement \( (A \cup B)^c \) is the elements in the universal set not in \( A \cup B \), which are \( \{j, k, l, m\} \).

Answer:

\(\{j, k, l, m\}\)