Thread: Abstract Algebra

Let A and B be sets and let f: A --> B be a function.
Define a relation on A as follows. If a,b in A, we say that aRb if and only if there exists some c in B such that f(a) = c and f(b) = c. Prove that R is and equivalence relation on A.

Ok, I know that i need to show the 3 properties. transitive, symmetry, and reflexsive. I also notice that f(a)=f(b). can i assume that a=b?

thanks

Originally Posted by Juancd08

Let A and B be sets and let f: A --> B be a function. Define a relation on A as follows. If a,b in A, we say that aRb if and only if there exists some c in B such that f(a) = c and f(b) = c. Prove that R is and equivalence relation on A. Can i assume that a=b?

Absolutely not. That would imply the f is injective.
However, this is an easy problem if one understands the set theoretical definition of functions.
The three necessary properties just ‘fall’ out of that definition.
For example, f(a)=f(a)=c means that aRa, reflexive.