Question

The sum of any two numbers and the sum obtained by changing the order of those numbers is the same'. How is this statement written using letters for number? 1d (1) a+b=x+y (3) axb=bxa (2) a + b =b+a (4) a+b=ba​

Answer 1

if a & b are two non- empty subsets and we have to prove axb=bxa iff a=b

proof:

we will prove this in two parts. in first part we will prove that if a=b then axb=bxa and in the second part we will prove that if axb=bxa then a=b.

i) for the first part let us assume that a=b

then in axb we can first replace first ‘a’ with ‘b’ (as by assumption a=b) so that it becomes bxb. now we have axb=bxb. in the next step we replace ‘b’ with ‘a’ so that bxb can be written as bxa.

thus we have axb=bxb=bxa

ii)for the second part we assume that axb=bxa and then we will prove that a=b. we will prove this by double containment. we will prove that a is subset of b and then b is subset of a .

let x∈ a and y∈b

now (x, y) ∈ axb

but by our assumption axb and bxa are equal

so (x,y) ∈ bxa implying that x ∈b , sine x is an arbitrarily chosen element, so a is subset of b.

now let x∈b and y∈a

so (x,y) ∈ bxa

but again since bxa=axb; therefore x∈a. thus implying as before that bis subset of a

thus from double containment viz. a being subset of b and b being subset of a; we get a=b.

Answer 2

hope the attachment will helps u..

thank u!!