Discrete math help! relations!?

Let A be the set of all ordered pairs of positive integers (including zero). This means that A = {(a, b)|a, b %26gt;= 0}.


Let R be a relation defined on A such that R = {[(a, b), (c, d)]|a+c = b+d}. Determine whether or not R is an


equivalence relation.





Let A be the set of all bit strings of length 25. Let R be the relation define on A where two bit strings are related


if the first 3 bits are the same. Show that R is an equivalence relation and enumerate one bit string from each


of the different equivalence classes of R.

Discrete math help! relations!?
Let's try the first. determine if R is reflexive, symmetric and transitive.





Is (a,b) R (a,b), ie, check that





a + a = b + b. True only if a = b, so, off the bat, this is NOT an equivalence relation!








As for the bit strings,


let s = abc12345..





Is it reflexive? Since s=s, yes.





Is it symmetric?





say abc1234...is related to abcsdfgsfg ....





Is abcsdfgsfg .... related to abc1234 ... yes, since the first three bits are same.





Is it transitive.





say abc123 ... is related to abc456... and abc456... is related to abc%26amp;*^... . Clearly, abc123 ... is related to abc%26amp;*^..., hence is an equivalent relations. Please note, I ignored the fact that 0 and 1 are the only allowed symbols.