Question

use the following venn diagram to find ( a^c cup b^c ).
venn diagram: two circles (a: purple with a, b, c; intersection: d, e; b: green with f, g, h, i; outside both: j, k, l, m). below: multiple-choice options:
{\\{a,b,c,f,g,h,i,j,k,l,m\\}
\\{a,g,i,l,r,s,t,u,v,w,x\\}
\\{a,b,d,f,g,j,l,o,s,x,y\\}
\\{f,h,j,o,q,s,u,v,w,y,z\\}
\\{b,c,f,l,m,q,s,t,v,y,z\\}
\\{\emptyset\\}

Explanation:

Step1: Recall De Morgan's Law

By De Morgan's Law, \( A^c \cup B^c=(A\cap B)^c \). Alternatively, we can find \( A^c \) (elements not in \( A \)) and \( B^c \) (elements not in \( B \)) and then take their union.

First, identify the universal set (the rectangle) and the sets \( A \) and \( B \). From the Venn diagram:

• Set \( A \) (purple and blue overlap): \( \{a, b, c, d, e\} \)
• Set \( B \) (green and blue overlap): \( \{d, e, f, g, h, i\} \)
• Elements outside both \( A \) and \( B \) (in the rectangle but not in \( A \) or \( B \)): \( \{j, k, l, m\} \)

Step2: Find \( A^c \)

\( A^c \) is all elements not in \( A \). So elements in \( B \) but not in \( A \) plus elements outside both: \( \{f, g, h, i, j, k, l, m\} \)

Step3: Find \( B^c \)

\( B^c \) is all elements not in \( B \). So elements in \( A \) but not in \( B \) plus elements outside both: \( \{a, b, c, j, k, l, m\} \)

Step4: Find \( A^c \cup B^c \)

Union of \( A^c \) and \( B^c \) is elements in \( A^c \) or \( B^c \). Combining \( \{f, g, h, i, j, k, l, m\} \) and \( \{a, b, c, j, k, l, m\} \), we get \( \{a, b, c, f, g, h, i, j, k, l, m\} \)

Answer:

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