Discrete-mathematics

Question

Consider the following statements

1) If f is one-to-one function from an infinite set A to itself then f must be onto.

2)  If f is one-to-one function from an finite set A to itself then f must be onto.

3) Power set of countably infinite set is countably infinite.

which of the following is true -

1) I only

2) I and II only

3) All are true.

4) II only

Comments

Is it 3)???

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Given answer is option 4)

Answer 1

option 1 -  suppose A has elements  1,2,3 .......now A is mapped one to one to A itself

lets consider mapping 1-1 , 2-2 , 3-3 ...every element has atleast one mapping which means its onto

option 2 - is the same

option 3 - when a set is given....the power set is taken from the given set with various combinations which we know...so if its countably infinite then power set will also be countably infinite

ANS - All are true

Comments

I differ on option 3 being true.

Cantor's Theorem states- If S is a countably infinite set, then 2^S (the power set) is uncountably infinite.

The right choice maybe-  I and II only.