Q1. What is the number of subsets of the empty set?
Solution:
The empty set has exactly one subset, which is itself (∅).
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Q2. What is the union of the sets {1, 2} and {2, 3}?
Solution:
The union of two sets includes all elements from both sets without duplicates. Thus, {1, 2} ∪ {2, 3} = {1, 2, 3}.
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Q3. The value of cos(tan^(-1)(x)) is:
Solution:
Using the right triangle definition, cos(tan^(-1)(x)) = adjacent/hypotenuse = 1/√(1+x^2).
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Q4. If R is a relation on the set {a, b, c} defined by R = {(a, b), (b, c)}, which property does R NOT have?
Solution:
R is not symmetric as (b,c) does not imply (c,b) is in R. It is reflexive and transitive.
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Q5. What is the union of the sets I = {1, 2} and J = {2, 3}?
Solution:
The union of two sets includes all unique elements from both sets. Thus, I ∪ J = {1, 2, 3}.
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Q6. If G = {x, y}, what is the number of subsets of G?
Solution:
The number of subsets of a set with n elements is 2^n. Here, n=2, so the number of subsets is 2^2 = 4.
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Q7. Find the value of cos^(-1)(0).
Solution:
cos^(-1)(0) = π/2, since cos(π/2) = 0.
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Q8. If F = {x, y, z}, what is the size of the power set of F?
Solution:
The size of the power set is 2^n where n is the number of elements in the set. Here, n = 3, so the size is 2^3 = 8.
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Q9. If F = {1, 2, 3}, what is the number of elements in the power set of F?
Solution:
The power set of a set with n elements has 2^n elements. For F, n = 3, so the power set has 2^3 = 8 elements.
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Q10. If f: A → B is a function and |A| = 5, |B| = 3, what is the maximum number of distinct functions f?
Solution:
The number of distinct functions from set A to set B is given by |B|^|A|. Here, |B| = 3 and |A| = 5, so the maximum number of distinct functions is 3^5 = 243.
