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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In a class of 35 students, 24 like to play cricket and 16 like to play football. Also, each student likes to play at least one of the two games. How many students like to play both cricket and football?
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.
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If A, B, C are three sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C, then:
Answer: B
This is a standard property of sets. Consider an element x ∈ B. Then x ∈ A ∪ B. Since A ∪ B = A ∪ C, we have x ∈ A ∪ C. So, x ∈ A or x ∈ C.
Case 1: x ∈ A. Then x ∈ A and x ∈ B, so x ∈ A ∩ B. Since A ∩ B = A ∩ C, we have x ∈ A ∩ C, which implies x ∈ C.
Case 2: x ∉ A. Since x ∈ A ∪ C, we must have x ∈ C.
In both cases, if x ∈ B, then x ∈ C. So, B ⊂ C. Similarly, we can prove that C ⊂ B. Therefore, B = C.
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The set {x | x is a real number and x² = -1} is a:
Answer: D
The equation is x² = -1. There is no real number whose square is negative. Therefore, the set contains no elements. A set with no elements is called an empty set or a null set (∅).
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In a school, there are 20 teachers who teach mathematics or physics. Of these, 12 teach mathematics and 4 teach both physics and mathematics. How many teach physics?
Answer: B
Let M be the set for Math teachers and P for Physics teachers.
n(M ∪ P) = 20, n(M) = 12, n(M ∩ P) = 4.
We use the formula n(M ∪ P) = n(M) + n(P) - n(M ∩ P).
20 = 12 + n(P) - 4
20 = 8 + n(P)
n(P) = 12.
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