CAT Previous Year Questions| Sets

Consider the sets A and B:

A = {62^n - 35^n - 1: n = 1, 2, 3, …}
B = {35(n - 1): n = 1, 2, 3, …}

Let's analyze the sets A and B:

A is defined by the expression 62^n - 35^n - 1, where n is a positive integer. It includes terms where both 62^n and 35^n are involved.

B is defined by the expression 35(n - 1), where n is a positive integer.

Now, let's analyze the statements:

A. Every member of A is in B and at least one member of B is not in A.
This statement is not true because A and B are defined differently, and not every member of A is in B.

B. At least one member of A is not in B.
This statement is true because the elements in A involve both 62^n and 35^n, while B only involves 35(n - 1). So, not all members of A are in B.

C. Every member of B is in A.
This statement is not true because B is a subset of A, and not all members of B are in A.

D. Neither every member of A is in B nor every member of B is in A.
This statement is not accurate because it's clear that A contains elements that are not in B, and B contains elements that are in A. So, it's not a "neither…nor" situation.

Therefore, the correct answer is (B) "At least one member of A is not in B."

To find the number of elements in (PΔQ)Δ(RΔS), let's break it down step by step.

1. Find PΔQ (the set of elements belonging to P or Q but not both):
   PΔQ = {1, 2, 3, 4} Δ {2, 3, 5, 6}
   PΔQ includes elements that belong to either P or Q but not both. So, it contains {1, 4, 5, 6}.
2. Find RΔS (the set of elements belonging to R or S but not both):
   RΔS = {1, 3, 7, 8, 9} Δ {2, 4, 9, 10}
   RΔS includes elements that belong to either R or S but not both. So, it contains {1, 3, 7, 8, 2, 4, 10}.
3. Now, find (PΔQ)Δ(RΔS) (the set of elements belonging to PΔQ or RΔS but not both):
   (PΔQ)Δ(RΔS) = {1, 4, 5, 6} Δ {1, 3, 7, 8, 2, 4, 10}
   (PΔQ)Δ(RΔS) includes elements that belong to either (PΔQ) or (RΔS) but not both. The combined set {1, 4, 5, 6} and {1, 3, 7, 8, 2, 4, 10} contains elements {5, 6, 3, 7, 8, 2, 10}.
4. Calculate the number of elements in the resulting set:
   The number of elements in (PΔQ)Δ(RΔS) is 7.

So, the correct answer is (D) 7.