defined by for all ...........it is not R "implies" R by the way, it is f:R --> R, or f maps from R to R, or f: R to R or something
i leave it to you to check that this is a reflexive, symmetric and transitive relation (aka, an equivalence relation)
do you know the definitions for one-to-one and onto?4. Suppose f: A implies B and g: B implies C are functions. If g o f is one to one and f is onto, show that g is one to one.
remember (g o f)(x) = g(f(x)). what does it mean for this function to be one-to-one? what does it mean for f to be onto? playing around with these definitions should help
what definitions are you working with? i'd say use induction on the cardinality of one of the sets. but maybe the definition you are using will give us an easier way. you may also want to look at the proof of theorem 1 here5. Let [A] = n and [b] = m for n,m belongs to N. use the definition of cardinality of a finite set to show that if A intersects B = empty set, then [A union B] = n + m