We will draw the Venn diagram to show the universal st U and the 3 sets A, B, C
Now we will start with the common elements of A and B
5 is the common element of A and B
1 is the common element between A and C
2, 4, and 6 are the common elements between B and C
3 is in A only
Now we can answer the question using the Venn diagram
1. A' means all the elements in U except A, then
[tex]A^{\prime}=\lbrace2,4,6\rbrace[/tex]
2. B U C means all elements in B and C
[tex]B\cup C=\lbrace1,2,4,5,6\rbrace[/tex]
3. A intersect C means common elements in A and C
[tex]A\cap C=\lbrace1\rbrace[/tex]
4. B' all elements in U not in B
[tex]B^{\prime}=\lbrace1,3\rbrace[/tex]
5. A U B all elements in A and B
[tex]A\cup B=\lbrace1,2,3,4,5,6\rbrace[/tex]
a. (A U C}' means all elements in U except A and C
[tex](A\cup C)^{\prime}=\lbrace\rbrace=\Phi[/tex]
Empty set because all elements are in A and C
b. A' intersects C'
[tex]\begin{gathered} A^{\prime}=\lbrace2,4,6),C^{\prime}=\lbrace3,5\rbrace \\ A^{\prime}\cap C^{\prime}=\lbrace\rbrace=\Phi \end{gathered}[/tex]
No common elements between A' and C'
c. (A intersects B)' means all elements in U except the common elements between A and B
[tex]\begin{gathered} A\cap B=\lbrace5\rbrace \\ \\ (A\cap B)^{\prime}=\lbrace1,2,3,4,6\rbrace \end{gathered}[/tex]
