Right then who can do this tuff cookie (2)

2. Let AxB be the set of ordered pairs (a,b) where a and b belong to the set of natural numbers N.

A relation p: AxB ----> AxB is defined by: (a,b)p(c,d) <-----> a+d = b+c

(i) Is (2,6) related to (4,8)? Give three ordered pairs which are related to (2,6)

(ii) Is (2,6) related to (a, a+4)? Justify your answer

(iii) Prove (i.e. using general letters) that p is an equivalence relation on the set AxB

**EXTENSION**

(iv) As p is an equivalence relation there are accociated equivalece classes. Find all the ordered pairs in the equivalence class of (2,6). Why could this equivalence class be identified with the integer -4?

(v) Give the equivalence classes (as sets of ordered pairs) defined by p for each of the integers: 0, -1 and +1

(vi) Consider two general ordered pairs, (a,b) and (c,d). If addition is defined by (a,b) + (c,d) = (a+c, b+d) and multiplication is defined by (a,b) x (c,d) = (ac+bd, ad+bc), show that these definitions provide a way of demonstrating that (+1) + (-1) = 0 and (-1) x (-1) = (+1)