Venn Diagrams and Set Notation

Set A is everything inside circle A, including the overlap.

The symbol \cap means the numbers that are in both sets.

The symbol \cup means the numbers in A or B or both.

A' means the numbers in \mathcal{E} that are not in A.

Check whether 15 belongs to set A, and then whether it belongs to set B.

These numbers are inside the rectangle but outside both circles.

1. Look for the numbers outside both circles.
2. They are 5, 11, 13 and 17.

Count every number inside circle B, including the overlap.

1. In B only: 24, 28, 30.
2. In the overlap: 22, 26.
3. 3+2=5 numbers are in set B.

Leave out the numbers in the overlap, because they are in B as well.

1. The numbers in A are 1, 2, 3, 6, 9.
2. 2 and 6 are in the overlap, so they are also in B.
3. In A but not in B: 1, 3, 9.

A number in the overlap must be a multiple of 3 and also odd.

1. Multiples of 3 up to 15: 3, 6, 9, 12, 15.
2. The odd ones are 3, 9 and 15.
3. So 3, 9 and 15 go in the overlap.

Count the numbers in circle B, and count all the numbers in the diagram.

1. Numbers in B: 6, 12, 18, 24. That is 4 numbers.
2. There are 12 numbers in the diagram altogether.
3. Probability =\frac{4}{12}=\frac{1}{3}.

Find the numbers in both sets first, then the rest of A, the rest of B, and finally the numbers in neither.

Part (a)

1. 24 and 30 are in both lists.

Part (b)

1. Take 24 and 30 out of A: 21, 27.

Part (c)

1. Take 24 and 30 out of B: 22, 26, 28.

Part (d)

1. The numbers in \mathcal{E} that are in neither list: 20, 23, 25, 29.

First think about which numbers are in A\cup B; the dash means everything else.

1. A\cup B is every number inside either circle: 12, 15, 16, 18, 20, 21.
2. (A\cup B)' is every number not in A\cup B, so the numbers outside both circles.
3. 11, 13, 14, 17, 19, 22.

All four numbers in the diagram add up to the number of students asked.

1. The numbers shown add up to 9+7+6=22.
2. There are 30 students altogether.
3. 30-22=8 students play both.

List the factors of 12 and the even numbers up to 12, then check where each number in the diagram should be.

1. Factors of 12: 1, 2, 3, 4, 6, 12. Even numbers: 2, 4, 6, 8, 10, 12.
2. 4 is a factor of 12 and it is even, so it belongs in the overlap, not in B only.
3. 9 is not a factor of 12 and it is odd, so it belongs outside both circles, not in A only.
4. The two numbers in the wrong place are 4 and 9.

B' means every number in the diagram that is not inside circle B.

1. Numbers not in B: 30, 33, 36 (in A only) and 35, 37 (outside both).
2. That is 5 numbers out of 11 in the diagram.
3. Probability =\frac{5}{11}.

List the factors of 18 and the multiples of 3 up to 18 first. Numbers on both lists go in the overlap.

Part (a)

1. Factors of 18: 1, 2, 3, 6, 9, 18.
2. Multiples of 3 up to 18: 3, 6, 9, 12, 15, 18.
3. On both lists: 3, 6, 9, 18.

Part (b)

1. Numbers in A or B or both: 1, 2, 3, 6, 9, 12, 15, 18. That is 8 numbers.
2. There are 18 numbers in \mathcal{E}.
3. 18-8=10 numbers go outside both circles (4, 5, 7, 8, 10, 11, 13, 14, 16, 17).

Part (c)

1. A\cup B is the numbers in A or B or both: 1, 2, 3, 6, 9, 12, 15, 18.
2. That is 8 numbers out of 18.
3. Probability =\frac{8}{18}=\frac{4}{9}.

Find A\cap B first. The dash means every other number in the diagram.

1. A\cap B is the overlap, which holds only 6.
2. (A\cap B)' is every number in \mathcal{E} except 6.
3. 1, 2, 3, 4, 5, 7, 8, 9.

Exactly one language means French only or Spanish only, but not both.

1. French only: 14. Spanish only: 11.
2. 14+11=25 students study exactly one language.
3. Probability =\frac{25}{40}=\frac{5}{8}.

A number in the middle, where all three circles overlap, is in every one of the sets.

Part (a)

1. A\cap B is the numbers inside both circle A and circle B.
2. 2 and 10 are in A and B only, and 4 is in all three circles.
3. 2, 4, 10.

Part (b)

1. B: 2, 4, 6, 10, 12, 14.
2. C: 4, 7, 9, 12.
3. Each number once: 2, 4, 6, 7, 9, 10, 12, 14.

Part (c)

1. The numbers in C are 4, 7, 9, 12: that is 4 numbers.
2. There are 15 numbers in \mathcal{E}.
3. Probability =\frac{4}{15}.

Put the 8 in the overlap first, then work out the other parts of each circle.

Part (a)

1. 26 went to art club and 8 of them also went to drama club.
2. 26-8=18.

Part (b)

1. Drama club only: 21-8=13.
2. Students inside the circles: 13+8+18=39.
3. Neither: 50-39=11.

Part (c)

1. Drama club only: 13 students.
2. Probability =\frac{13}{50}.