Venn Diagrams

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SAT Math › Venn Diagrams

Questions 1 - 10

Venn 1

In the above Venn diagram, let the universal set be

$U =\left { 1, 2, 3, ...40 \right }$

$A = \left { x | x$ yields a remainder of 1 when divided by 4 $\left \right }$

$B = \left { x | x$ yields a remainder of 1 when divided by 3 $\left \right }$

How many elements of would be placed in the shaded portion of the above diagram?

Six

Seven

Eight

Nine

Five

Explanation

The shaded portion of the Venn diagram is $A \cap B'$ - the set of all elements in but not .

The following elements of yield a remainder of 1 when divided by 4, and therefore comprise set :

$A = \left { 1, 5, 9, 13, 17, 21, 25, 29, 33, 37 \right }$

If 3 is subtracted from each, what results are the elements whose division by 4 yields a remainder of 1, and thus, elements of .

The following elements of yield a remainder of 1 when divided by 4, and therefore comprise set :

$B = \left { 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40 \right }$

The elements in that are also elements in are 1, 13, 25, and 37 - four elements out of ten. Therefore, the set $A \cap B'$ comprises six elements.

Venn 1

In the above Venn diagram, let the universal set be

$U =\left { 1, 2, 3, ...40 \right }$

$A = \left { x | x$ yields a remainder of 1 when divided by 4 $\left \right }$

$B = \left { x | x$ yields a remainder of 1 when divided by 3 $\left \right }$

How many elements of would be placed in the shaded portion of the above diagram?

Six

Seven

Eight

Nine

Five

Explanation

The shaded portion of the Venn diagram is $A \cap B'$ - the set of all elements in but not .

The following elements of yield a remainder of 1 when divided by 4, and therefore comprise set :

$A = \left { 1, 5, 9, 13, 17, 21, 25, 29, 33, 37 \right }$

If 3 is subtracted from each, what results are the elements whose division by 4 yields a remainder of 1, and thus, elements of .

The following elements of yield a remainder of 1 when divided by 4, and therefore comprise set :

$B = \left { 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40 \right }$

The elements in that are also elements in are 1, 13, 25, and 37 - four elements out of ten. Therefore, the set $A \cap B'$ comprises six elements.

Venn 1

In the above Venn diagram, let the universal set be

$U =\left { 1, 2, 3, ...40 \right }$

$A = \left { x | x$ yields a remainder of 1 when divided by 4 $\left \right }$

$B = \left { x | x$ yields a remainder of 1 when divided by 3 $\left \right }$

How many elements of would be placed in the shaded portion of the above diagram?

Six

Seven

Eight

Nine

Five

Explanation

The shaded portion of the Venn diagram is $A \cap B'$ - the set of all elements in but not .

The following elements of yield a remainder of 1 when divided by 4, and therefore comprise set :

$A = \left { 1, 5, 9, 13, 17, 21, 25, 29, 33, 37 \right }$

If 3 is subtracted from each, what results are the elements whose division by 4 yields a remainder of 1, and thus, elements of .

The following elements of yield a remainder of 1 when divided by 4, and therefore comprise set :

$B = \left { 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40 \right }$

The elements in that are also elements in are 1, 13, 25, and 37 - four elements out of ten. Therefore, the set $A \cap B'$ comprises six elements.

$Set\ A=\left { 2,4,6,8,10,12 \right }$ and $Set\ B=\left { 5,6,7,8,9 \right }$ .

Find $A\bigcap B$ .

$\left { 6,8 \right }$

$\left { 6,9 \right }$

$\left { \ \right }$

$\left {5,7,9 \right }$

$\left {6,7,8 \right }$

Explanation

The intersection of two sets contains every element that is present in both sets, so $\left { 6,8 \right }$ is the correct answer.

$Set\ A=\left { 2,4,6,8,10,12 \right }$ and $Set\ B=\left { 5,6,7,8,9 \right }$ .

Find $A\bigcap B$ .

$\left { 6,8 \right }$

$\left { 6,9 \right }$

$\left { \ \right }$

$\left {5,7,9 \right }$

$\left {6,7,8 \right }$

Explanation

The intersection of two sets contains every element that is present in both sets, so $\left { 6,8 \right }$ is the correct answer.

$Set\ A=\left { 2,4,6,8,10,12 \right }$ and $Set\ B=\left { 5,6,7,8,9 \right }$ .

Find $A\bigcap B$ .

$\left { 6,8 \right }$

$\left { 6,9 \right }$

$\left { \ \right }$

$\left {5,7,9 \right }$

$\left {6,7,8 \right }$

Explanation

The intersection of two sets contains every element that is present in both sets, so $\left { 6,8 \right }$ is the correct answer.

What is the intersection of the Venn Diagram shown below?

Screen shot 2015 10 27 at 3.45.31 pm

Explanation

The intersection of the Venn Diagram is only the numbers in both circles.

The section in the middle contains the answer set.

Thus the intersection is, .

We have two sports clubs offered to a class of 100 students. 70 students joined the basketball club, 40 students joined the swimming club, and 10 students joined neither. How many students joined both the swimming club and the basketball club?

Explanation

The idea is to draw a Venn Diagram and find the intersection. We have one circle of 70 and another with 40. When we add the two circles plus the 10 students who joined neither, we should get 100 students. However, when adding the two circles, we are adding the intersections twice, therefore we need to subtract the intersection once.

We get , which means the intersection is 20.

Set A contains the positive even integers less than 14. Set B contains the positive multiples of three less than 20. What is the intersection of the two sets?

A∩B = { }

A∩B = {6, 12}

A∩B = {4, 6, 8}

A∩B = {6}

A∩B = {6, 12, 18}