Prove 3k k elementof 4 7k k elementof 4SolutionS3k k in Z

Prove: {3k: k elementof 4} = {7k = k elementof 4}

Solution

S={3k: k in Z}

T={7k: k in Z}

Define map f from S to T

f(x)=7x/3

Let, f(x)=f(y)

SO,

7x/3=7y/3

Hence, x=y

Let, y be in T

So,

f(3y/7) =y        (Note 3y/7 is an integer because y is in T hence y is multiple of 7)

Hence, f is onto

HEnce f is a bijection from S to T

HEnce, |S|=|T|