1 A. B. C. D. E. Question 2 A. B. C. D. E. True or False, Please answer if the statement if True or False and give a detailed reply as to WHY!!! 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. True or false. A set is any collection of objects. True or false. A proper subset of a set is itself a subset of the set, but not vice versa. True or false. The empty set is a subset of every set. True or false. If A B = , then A = and B = . True or false. If A B = , then A = or B = or both A and B are empty sets. True or false. (A Ac)c = . True or false. [A (B C)]c = (A B)c (A C)c True or false. n(A) + n(B) = n(A B) + n(A B) True or false. If A B, then n(B) = n(A) + n(Ac B). True or false. The number of permutations of n distinct objects taken all together is n! True or false. P(n, r) = r! C(n, r). Question1 B. Question 2 B. True or False. True. A set is a collection of objects. The objects are referred to as elements of the set. If S is a set and x is some element in S, we write x S. 1. . False. A "proper subset" of a set A is defined as a set B that is contained by A, but is not equal to A. If A had a subset B, where B is defined as A, then A = B, and thus does not satisfy the conditions for a PROPER subset, although it is still always subset of itself. 2. . True. An empty set contains no elements while a subset contains elements that are not in the other comparing set. Hence an empty set becomes a subset of all the other sets because it has no elements and the other set contains elements. 3. . True. We see that the union of both sets presents an empty set. So this can happen only if A and B, both sets are empty sets. 4. . False. If the intersection of two states is an empty set, it does not mean that any of the sets is empty. It also means that there are no common elements in both A and B and these sets may not be empty. 5. . True. AC means compliment of set A i.e. all elements leaving the elements of set A. So, the union of A and AC will be U (universal set) i.e. all the elements are included. Taking the compliment of the result will give a null set. Excluding the Universal, there are no elements. All the elements are incorporated in the universal set. 6. . True. Let A={a , b , c} , B = {c , d , e} and C = {b , e , f} U = {a,b,c,d,e,f} B C = { b,c,d,e,f} A (B C) = {b,c} [A (B C)]c = {a,d,e,f} A B = {c} (A B)c = {a,b,d,e,f} A C = {b} (A C)c ={a,c,d,e,f} (A B)c (A C)c = {a,d,e,f} So, [A (B C)]c = (A B)c (A C)c 7. . True. The n(A)+n(B) includes the number of elements common in both which is added twice. If we have 5 elements in set A and 4 elements in set B out of which 2 elements are common in both the sets, So, n(A)+n(B) = 9 n(AUB) = 7 as only two elements are common in both the sets. n(A B) = 2. So, n(A) + n(B) = n(A B) + n(A B) 8. . False. Suppose there are 5 elements in set A and 4 elements in set B. There are a total of 10 elements in the Universal set. So, n(B) = 4, n(A) = 5. AC will have 5 elements out of which 4 are there in set B. So, n(Ac B) = 4 So, n(B) n(A) + n(Ac B). 9. . True. A permutation of n objects is an ordered arrangement of the n objects. This means we need to arrange all n objects at a time which means P(n,r) = n!/(n-n)! = n! 10. . True. C(n, r) = And P(n, r) = n! r ! ( nr ) ! n! ( nr ) ! So, P(n, r) = r! C(n, r)