If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, what is A ∪ B?
Answer: B
The union of two sets, denoted by A ∪ B, is the set of all elements that are in A, or in B, or in both.
A = {1, 2, 3, 4}
B = {3, 4, 5, 6}
A ∪ B = {1, 2, 3, 4, 5, 6}
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If A = {a, b, c, d} and B = {c, d, e, f}, what is A ∩ B?
Answer: C
The intersection of two sets, denoted by A ∩ B, is the set of all elements that are common to both A and B.
A = {a, b, c, d}
B = {c, d, e, f}
A ∩ B = {c, d}
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In a school, 300 students play cricket and 250 play hockey. If 110 students play both games, how many students play either cricket or hockey?
Answer: B
Let C be the set for cricket and H for hockey.
n(C) = 300, n(H) = 250, n(C ∩ H) = 110.
The number of students who play either game is the union, n(C ∪ H).
n(C ∪ H) = n(C) + n(H) - n(C ∩ H)
n(C ∪ H) = 300 + 250 - 110 = 550 - 110 = 440.
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The Idempotent Law for set union states that:
Answer: A
The Idempotent Law states that when an operation is applied to an element with itself, the result is the same element.
For set theory, this means:
1. A ∪ A = A (Idempotent Law of Union)
2. A ∩ A = A (Idempotent Law of Intersection)
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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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If n(A) = 115, n(B) = 326, n(A-B) = 47, then what is n(A ∪ B) equal to?
Answer: A
We know that n(A) = n(A-B) + n(A ∩ B).
115 = 47 + n(A ∩ B)
n(A ∩ B) = 115 - 47 = 68.
Now we find the union:
n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
n(A ∪ B) = 115 + 326 - 68 = 441 - 68 = 373.
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In a competition, a school awarded medals in different categories. 36 medals in dance, 12 in dramatics and 18 in music. If these medals went to a total of 45 persons and only 4 persons got medals in all the three categories, how many received medals in exactly two of these categories?
Answer: C
Let D, R, M be the sets for Dance, Dramatics, Music. n(D)=36, n(R)=12, n(M)=18. n(D∪R∪M)=45. n(D∩R∩M)=4. Let x be the number of people who received medals in exactly two categories.
We know n(D∪R∪M) = (Sum of exactly one) + (Sum of exactly two) + (Sum of exactly three). Also, n(D∪R∪M) = n(D)+n(R)+n(M) - (Sum of pairs) + n(D∩R∩M). 45 = 36+12+18 - (Sum of pairs) + 4 => 45 = 70 - (Sum of pairs). So Sum of pairs = 25.
We know (Sum of pairs) = (Exactly two) + 3 * (Exactly three). 25 = x + 3 * 4 => 25 = x + 12 => x = 13. Wait, let me re-check. Yes, Sum of n(A∩B) = Sum of exactly two + 3 * n(A∩B∩C). That gives x=13. Why is the answer 11? Let me try another formula. (Sum of exactly one) = (n(A)+..) - 2(Sum of pairs) + 3(All three). Let's use the basic one. 45 = (n(D)+n(R)+n(M)) - (n(D∩R)+..) + n(D∩R∩M) => 45 = 66 - (n(D∩R)+..) + 4 => Sum of intersections = 25. This is correct. Let's use the formula for total people = (only A + only B + only C) + (exactly two) + (exactly three). Let 'Exactly two' be x. Then 45 = (Only D+..+Only M) + x + 4. Only D = 36 - n(D∩R) - n(D∩M) + n(D∩R∩M). This becomes circular. Let's re-verify: sum of pairs = 25. Exactly two = 25 - 3*4 = 13. I am consistently getting 13. I will modify the answer to be 13.
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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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De Morgan's Law states that (A ∪ B)' is equal to:
Answer: B
De Morgan's Laws are fundamental principles in set theory.
The first law states that the complement of the union of two sets is the intersection of their complements: (A ∪ B)' = A' ∩ B'.
The second law states that the complement of the intersection of two sets is the union of their complements: (A ∩ B)' = A' ∪ B'.
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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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