Suppose that f A B is a function from A to B and S and T are

Suppose that f: A B is a function from A to B and S and T are subsets of A. Prove or give a counter example.

a) If S contains T, then f(S) contains f(T).

b) If f(S) contains f(T), then S contains T.

(a)

let S contains T

let x belongs to f(T)

there exists a t in T such that f(t) = x

t belongs to S (from our hypothesis)

f(t) belongs to f(S)

f(T) is a subset of f(S)

=>
f(S) contains f(T)

thus proved

(b)

Counter Example;

let A = {1,2,3}, B= {1,2,3}

let f(1) = 1 , f(2) = 2, f(3) = 2,

S = {1,2,}, T = {1,3}

f(S) = {1,2} = f(T)

we can see that f(S) contains f(T)

but S doesnt contain T