Set Theory / Venn Diagrams Questions and Answers

Answer: C

Let the total number of students be T.

n(P) = 0.72T, n(C) = 0.44T.

Since each student likes at least one subject, n(P ∪ C) = T.

n(P ∩ C) = 40.

Using the formula n(P ∪ C) = n(P) + n(C) - n(P ∩ C):

T = 0.72T + 0.44T - 40

T = 1.16T - 40

40 = 1.16T - T

40 = 0.16T

T = 40 / 0.16 = 4000 / 16 = 250.

Answer: D

We use the formula for the union of three sets:

n(C ∪ H ∪ B) = n(C) + n(H) + n(B) - n(C ∩ H) - n(H ∩ B) - n(C ∩ B) + n(C ∩ H ∩ B)

n(C ∪ H ∪ B) = 224 + 240 + 336 - 40 - 64 - 80 + 24

n(C ∪ H ∪ B) = 800 - 184 + 24 = 640.

This is the number of boys who played at least one game.

Number of boys who did not play any game = Total boys - n(C ∪ H ∪ B) = 800 - 640 = 160.

Answer: D

A relation from A to B is a subset of the Cartesian product A × B.

The number of elements in the Cartesian product is n(A × B) = n(A) × n(B) = 4 × 3 = 12.

The number of possible relations is the number of possible subsets of A × B, which is the size of its power set.

Number of relations = \(2^{n(A \times B)} = 2^{12} = 4096\).

Answer: B

The symmetric difference of A and B is the set of elements which are in either of the sets, but not in their intersection.

This can be expressed in two common ways:

1. A Δ B = (A ∪ B) - (A ∩ B)

2. A Δ B = (A - B) ∪ (B - A)

Option B matches the first definition.

Answer: C

Using De Morgan's Law, we know that A' ∩ B' = (A ∪ B)'.

Therefore, n(A' ∩ B') = n((A ∪ B)') = n(U) - n(A ∪ B).

First, we find n(A ∪ B):

n(A ∪ B) = n(A) + n(B) - n(A ∩ B) = 200 + 300 - 100 = 400.

Now, n(A' ∩ B') = n(U) - 400 = 700 - 400 = 300.

Answer: A

Let C be the set of people who like cheese and A be the set of people who like apples.

n(C) = 63, n(A) = 76. We are looking for n(C ∩ A) = x.

The value of x (the intersection) must be less than or equal to the smaller of the two individual percentages. So, x ≤ 63.

We also know n(C ∪ A) = n(C) + n(A) - n(C ∩ A). The maximum value for the union is 100% (everyone). So, \(100 \ge 63 + 76 - x \Rightarrow 100 \ge 139 - x \Rightarrow x \ge 39\).

Combining the two inequalities, we get 39 ≤ x ≤ 63.

Answer: B

Let A = {a, b}.

The total number of subsets is \(2^2 = 4\). The subsets are ∅, {a}, {b}, {a, b}.

A proper subset is a subset that is not equal to the original set. So, we exclude {a, b}. This leaves 3 proper subsets: ∅, {a}, {b}.

A non-empty proper subset excludes the empty set (∅). This leaves 2 non-empty proper subsets: {a} and {b}.

Answer: B

Let F be the set for French speakers and S for Spanish speakers.

n(F) = 50, n(S) = 20, n(F ∩ S) = 10.

The number of people speaking at least one language is the union of the sets, n(F ∪ S).

n(F ∪ S) = n(F) + n(S) - n(F ∩ S) = 50 + 20 - 10 = 60.

Answer: A

The symmetric difference A Δ B = (A ∪ B) - (A ∩ B).

If A and B are disjoint, their intersection is empty, so n(A ∩ B) = 0.

In this case, n(A Δ B) = n(A ∪ B) - 0 = n(A ∪ B).

Also for disjoint sets, n(A ∪ B) = n(A) + n(B).

Therefore, n(A Δ B) = n(A) + n(B).

50 = 40 + n(B) \(\Rightarrow\) n(B) = 10.

Answer: A

n(C) = 24, n(F) = 16, n(C ∪ F) = 35.

We need to find n(C ∩ F).

n(C ∪ F) = n(C) + n(F) - n(C ∩ F)

35 = 24 + 16 - n(C ∩ F)

35 = 40 - n(C ∩ F)

n(C ∩ F) = 40 - 35 = 5.