Function Equivalence: Proving Equality of Functions in F(S,F)

[Seacow1988]

Homework Statement

Two functions in F(S,F) (or going from the vector space S to the vector space F) are equal if and only if they have the same value at each element of S. True or False?

Homework Equations

How can you prove: if two functions, x and y, are equal then they have the same value at each element of S?

The Attempt at a Solution

By the definition of equivalence, I can see that if two functions have the same value at each element of S, they are equal. However, I'm not sure how to show the converse.

 

[HallsofIvy]

Homework Statement

Two functions in F(S,F) (or going from the vector space S to the vector space F) are equal if and only if they have the same value at each element of S. True or False?

Homework Equations

How can you prove: if two functions, x and y, are equal then they have the same value at each element of S?

The Attempt at a Solution

By the definition of equivalence, I can see that if two functions have the same value at each element of S, they are equal. However, I'm not sure how to show the converse.

What is the definition of "equal" for functions? Typically, two mathematical "objects" are said to be "equal" if all parts of them are the same. Okay, what is the definition of "function"? What "parts" does a function have?