If A = {1, 2, {3, 4}}, which of the following is a true statement?
Answer: B
The elements of set A are 1, 2, and the set {3, 4}.
Option A is false because 3 is not an element of A; it's an element of the element {3, 4}.
Option B is true because 1 and 2 are elements of A, so the set containing them, {1, 2}, is a subset of A.
Option C is false because {3, 4} is an element of A, not a subset. For it to be a subset, 3 and 4 would have to be elements of A themselves.
Option D is false for the same reason as A.
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In a class, 72% students like Physics and 44% students like Chemistry. If each student likes at least one subject and there are 40 students who like both, what is the total number of students in the class?
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.
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Out of 800 boys in a school, 224 played cricket, 240 played hockey and 336 played basketball. Of the total, 64 played both basketball and hockey; 80 played cricket and basketball and 40 played cricket and hockey; 24 played all the three games. The number of boys who did not play any game is:
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.
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If n(A) = 4 and n(B) = 3, what is the number of possible relations from A to B?
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 = \(2^{n(A \times B)} = 2^{12} = 4096\).
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Let A = {x : x is an integer and -3 < x ≤ 4}. Find n(A).
Answer: B
The condition -3 < x ≤ 4 means x can be any integer strictly greater than -3 and less than or equal to 4.
The integers satisfying this condition are: -2, -1, 0, 1, 2, 3, 4.
The set A = {-2, -1, 0, 1, 2, 3, 4}.
The number of elements, n(A), is 7.
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If n(U) = 700, n(A) = 200, n(B) = 300 and n(A ∩ B) = 100, then n(A' ∩ B') is:
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.
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In a survey, 63% of Americans like cheese, whereas 76% like apples. If x% of the Americans like both cheese and apples, then find the range of x.
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.
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If a set has 2 elements, how many non-empty proper subsets does it have?
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}.
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In a committee, 50 people speak French, 20 speak Spanish and 10 speak both Spanish and French. The number of persons speaking at least one of these two languages is:
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.
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If n(A Δ B) = 50, n(A) = 40, find n(B) if A and B are disjoint.
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.
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