DM - Functions & Relations -Q15

+1 vote
if A is a set(finite or infinite) and P(A) is its power set, i.e. the set of all subsets of A, then consider the following statements :

A. There is No surjective function from A to P(A).

B. There is No injective function from A to P(A).

C. There is No surjective function from P(A) to A.

D. There is No injective function from P(A) to A.

Which of the above is Necessarily true ?

(a) . Only B,C

(b). Only A,D

(c). Only A,C

(d). Only B,D
asked Jun 24 in Discrete Maths by gbeditor (44,500 points)
reshown 6 days ago by gbeditor

1 Answer

0 votes
 
Best answer

Cantor's theorem shows that

There is No surjective function from A to P(A).

And

There is No injective function from P(A) to A.

For finite sets, we can easily see that these are true because if |A| = n then |P(A)| = 2^n

So, There is No surjective function possible from A to P(A).

And There is No injective function from P(A) to A.

https://www.whitman.edu/mathematics/higher_math_online/section04.10.html

answered 4 days ago by deepak-gatebook (112,390 points)
Answer:
...