Prove or disprove Let A x epsilon Zx 10a 3 for some integ

Prove or disprove: Let A = {x epsilon Z|x = 10a - 3 for some integer a} and B = {y epsilon Z|y = 10b + 7 for some integer b}. Then A = B.

Solution

Let, x be in A

x=10a-3 for som integer a,

x=10(a-1+1)-3=10(a-1)+7

Hence, x is in B

Let, y be in B

y=10b+7 for some integer b

y=10(b+1-1)+7=10(b+1)-3

HEnce, y is in A

Hence, A=B