Define sets A and B as follows A n Z n 8r 3 for some int

Define sets A and B as follows:

A = { n Z | n = 8r 3 for some integer r}

and

B = {m Z | m = 4s + 1 for some integer s}.

(a) Is A B?

(b) Is B A?

Justify your answers carefully. (In other words, provide a proof if the statement is true or a disproof if the statement is false.)

Discrete Math, Thank you!

Solution

(a)TRUE

let n belongs to A

=>

n = 8r-3 for somr integer 3

=>

n = 8(r-1+1) -3 = 8(r-1) +5 = 8(r-1) + 4 +1 = 4(2r-1) +1

=>
n belongs to B

=>

A is a subset of B

thus proved

(b) FALSE

counter example:

9 belongs to B but 9 doesnt belong to A