Set Theory, Calendars, Clocks and Binomial Theorem

You are here: Home  CAT Questionbank   CAT Quant  Set Theory  Question 12

Bijective Functions

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

 

  • Correct Answer
    Choice A. Yes

Detailed Solution

A/Q the functions f:A → B and g:C → D are bijections
Then g-1 must exist .
Then for a function h∈ Ac we may define a function T: AC → BBD by T(h) = f o h o 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 o h o g-1 is a bijection.
To do so, we need to simply provide an inverse.
Now T o R(h) = f o (f-1 o h o g) o g-1
= (f o f-1) o h o (g o g-1 )
= idB o h o idD
= h Therefore R:h → f-1 oh o g exists and is an inverse to T
Hence there exists a bijection between AC + BD
Correct Answer: Yes.



Our Online Course, Now on Google Playstore!

2IIM's App

Fully Functional Course on Mobile

All features of the online course, including the classes, discussion board, quizes and more, on a mobile platform.


Cache Content for Offline Viewing

Download videos onto your mobile so you can learn on the fly, even when the network gets choppy!

Get it on Google Play
Visit Piverb.com
Visit Wizako.com