SAT Sets and Venn Diagrams

To successfully solve problems about sets and Venn diagrams on the SAT, you need to know:

• the definitions of elements, sets, and subsets
• the meaning of union and intersection of sets
• how to work with Venn diagrams

Set $A = \{2, 3, 5, 8, 9\}$
Set $B = \{5, 8, 11\}$

How many elements in set $A$ are also in set $B?$

(A) $\ \$ Zero
(B) $\ \$ One
(C) $\ \$ Two
(D) $\ \$ Three
(E) $\ \$ Five

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Correct Answer: C

Solution:

It is easy to see that two elements that are in set $A$ are also in set $B:$ 5 and 8.

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Incorrect Choices:

(A) and (B)
These choices are just offered to confuse you.

(D)
This is how many elements in $A$ are not in $B,$ and also how many elements there are in set $B.$

(E)
This is how many elements there are in set $A.$

Sets $A, B,$ and $C$ are shown in the Venn diagram above. Each number indicates the number of elements in that region. How many elements are included in sets $B$ or $C$ but are not included in set $A?$

(A) $\ \ 6$
(B) $\ \ 7$
(C) $\ \ 11$
(D) $\ \ 17$
(E) $\ \ 26$

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Correct Answer: D

Solution:

As shown in the diagram above, there are 9 elements included in $B$ only, 2 elements included in $C$ only, and 6 elements included in both $B$ and $C,$ but not in $A.$ Therefore, there are 9+2+6=17 such elements.

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Incorrect Choices:

(A)
This is how many elements are common to both $B$ and $C$ that are not also included in $A.$

(B)
This is how many elements are common to both $B$ and $C.$

(C)
This is how many elements are in set $B$ only or set $C$ only.

(E)
This is how many elements are in $B$ or in $C,$ or in both $B$ and $C$.

If you thought these examples difficult and you need to review the material, these links will help:

• Sets
• Sets-Elements
• Subsets
• Unions and Intersections
• Sets-Problem Solving
• Venn Diagrams

• The union of two sets, $A$ and $B,$ is that collection of elements that are in $A,$ or in $B,$ or in both $A$ and $B.$
• The intersection of two sets, $A$ and $B,$ is that collection of elements that are only in both $A$ and $B.$
• If every element in set $A$ is an element in set $B,$ then $A$ is a subset of $B.$
• SAT General Tips