To prove is a transitive relation, you must show that if (a,b) and (b,c) are in B then (a,c) is also in B. That should be easy. If (a,b) and (b,c) are in B, then they are in EVERY relation in A, each of which is transitive: that means (a,c) is in every relation in A.

To prove that is a transitive relation you must show that if (a,b) and (b,c) are in C then (a,c) is also in C. If (a,b) and (b,c) are in C, then (a,b) is in SOME relation in A, (b,c) is in SOME relation in A, but not necessarily the same one!

A function is a set of ordered pairs such that no two distinct pairs have the same second member (in other words, you cannot have two pairs having different first members but the same second member).Also one more problem:

Let be a set of functions such that for anyfand anygin , either or . Show that is a function.

I think indirect proof is best here. Suppose there were two pairs, say (a, b) and (c, b) having the same second member but distinct first members. Since every member of A is a function, that could only happen if (a,b) were in one function, say f, and (c, b) were in another, say g. But if either or one cannot contain a pair that is not in the other.

I'd really appreciate any help,

Many thanks steph[/QUOTE]