CAT Quantitative Aptitude Questions | CAT Set theory Questions - Union and Intersection

The question is from CAT Set theory. It combines function with set theory. We need to determine type of a function between two sets. CAT exam is known to test on basics rather than high funda ideas. CAT also tests multiple ideas in the same question, and 2IIMs CAT question bank provides you with CAT questions that can help you gear for CAT Exam CAT 2023. Set Theory (especially constructing venn diagrams) is a frequently tested topic. Make sure you know the basics from this chapter.

Question 12: If set A and set B are bijective and set C and set D are bijective too, State whether there exist a bijection between A^C + B^D or not

1. Yes
2. No
3. Data insufficient
4. Cannot be determined

Explanatory Answer

Method of solving this CAT Question from Set theory: How bijective functions are defined?

\\frac{A}{Q}\\) the functions f:A → B and g:C → D are bijections
Then g^-1 must exist .
Then for a function h∈ A^c we may define a function T: A^C → BB^D by T(h) = f ∘ h ∘ g^-1 . That is , for d ∈ D, T(h)(d) = f(h(g^-1(d)))
Since g^-1:D→C, the expression g^-1(d) must exist

Now,
As h:C → A and g^-1(d) ∈ C then the expression h(g^-1(d)) must exist
Again, As h(g^-1(d)) ∈ A and f:A → B, the expression f(h(g^-1(d))) must exist

Now it only remains to prove that R(h) = f ∘ h ∘ g^-1 is a bijection.
To do so, we need to simply provide an inverse.

Now T ∘ R(h) = f ∘ (f^-1 ∘ h ∘ g) ∘ g^-1
= (f ∘ f^-1) ∘ h ∘ (g ∘ g^-1 )
= id_B ∘ h ∘ id_D
= h
Therefore R:h → f^-1 ∘ h ∘ g exists and is an inverse to T
Hence there exists a bijection between A^C + B^D

The question is "If set A and set B are bijective and set C and set D are bijective too, State whether there exist a bijection between AC + BD or not"

Hence, the answer is "Yes".

Choice A is the correct answer.