ISEE Upper Level Quantitative Reasoning - How to use a Venn Diagram

Venn

In the above Venn diagram, the universal set is defined as $U = \left { a, b, c, d, e, f, g, h\right }$ . Each of the eight letters is placed in its correct region. Which of the following is equal to $\overline{A\cup B}$ ?

$\overline{A\cup B} = \left { c\right }$

$\overline{A\cup B} = \left { a,c,f\right }$

$\overline{A\cup B} = \left { a,f\right }$

$\overline{A\cup B} = \left { b,c,d,e,g,h\right }$

$\overline{A\cup B} = \left { b,d,e,g,h\right }$

Explanation

$\overline{A\cup B}$ is the complement of $A\cup B$ - the set of all elements in not in $A\cup B$ .

$A\cup B$ is the union of sets and - the set of all elements in either or . Therefore, $\overline{A\cup B}$ is the set of all elements in neither nor , which can be seen from the diagram to be only one element - . Therefore,

$\overline{A\cup B} = \left { c\right }$

Venn

In the above Venn diagram, the universal set is defined as $U = \left { a, b, c, d, e, f, g, h\right }$ . Each of the eight letters is placed in its correct region.

What is $A \cap B$ ?

$A \cap B = \left { a,f \right }$

$A \cap B = \left { a,b,d,e,f,g,h \right }$

$A \cap B = \left { b,d,e,g,h \right }$

$A \cap B = \left { b,c,d,e,g,h \right }$

$A \cap B = \left { a,c,f \right }$

Explanation

$A \cap B$ is the intersection of sets and - that is, the set of all elements of that are elements of both and . We want all of the letters that fall in both circles, which from the diagram can be seen to be and . Therefore,

$A \cap B = \left { a,f \right }$

Venn

$\overline{A\cap B} = \left { b,c,d,e,g,h\right }$

$\overline{A\cap B} = \left { b,d,e,g,h\right }$

$\overline{A\cap B} = \left { c\right }$

$\overline{A\cap B} = \left { a,c,f\right }$

$\overline{A\cap B} = \left { a,f\right }$

Explanation

$\overline{A \cap B}$ is the complement of $A \cap B$ - the set of all elements in not in $A \cap B$ .

$A \cap B$ is the intersection of sets and - that is, the set of all elements of that are elements of both and . Therefore, $\overline{A \cap B}$ is the set of all elements that are not in both and , which can be seen from the diagram to be all elements except and . Therefore,

$\overline{A\cap B} = \left { b,c,d,e,g,h\right }$ .

The following Venn diagram depicts the number of students who play hockey, football, and baseball. How many students play football and baseball?

Problem_9

Explanation

The number of students who play football or baseball can by finding the summer of the number of students who play football alone, baseball alone, baseball and football, and all three sports.

A class of students was asked whether they have cats, dogs, or both.The results are depicted in the following Venn diagram. How many students do not have a dog?

Question_5

Explanation

First, calculate the number of students with a dog:

Next, subtract the number of students with a dog from the total number of students.

Venn

Refer to the above Venn diagram.

Define universal set $U = \mathbb{N}$ , the set of natural numbers.

Define sets and as follows:

$A = \left { 1,5,9,13,17,21,... \right }$

$B = \left { 2,5,8,11,14,17...\right }$

Which of the following numbers is an element of the set represented by the gray area in the diagram?

Explanation

The gray area represents the set of all elements that are in but not in .

is the set of integers that, when divided by 3, yield remainder 2. Therefore, we can eliminate 102 and 105, both multiples of 3, and 103, which, when divided by 3, yields remainder 1.

is the set of integers that, when divided by 4, yield remainder 1. Since we do not want an element from this set, we can eliminate 101, but not 104.

104 is the correct choice.

How to use a Venn Diagram

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