# What defines an operator input/output for simple expressions

1. Jan 5, 2013

### benno

Hello. I have some questions on operations.

Suppose in the course of a derivation there is a mathematical statement of the form
A+1=B+C
then "+" is an operator acting on inputs "B" and "C".
Question 1: Is the output of the operation "A" or the expression "B+C"?
The reason I think the output may be "B+C" is because in the statement X=(Y+Z)+W, wouldn't the output of the first addition be Y+Z?

Question 2: Since the equals sign "=" has two inputs, the left hand side (LHS) and right hand side (RHS) expressions, is it also an operator?
How does one distinguish whether A+1 is the input or B+C is the input?
If "=" is an operator, what is the output? If "=" is not an operator, what is it?

2. Jan 5, 2013

### I like Serena

Welcome to PF, benno!

The operation "+" on B and C has as output indeed "B+C".

The symbol "=" represents a "relation".

Suppose we're only talking about the real numbers $\mathbb R$ with the regular addition "+" and the regular equality "=".

Then, mathematically, "+" is a function, defined as:
$$+: \mathbb R \times \mathbb R \to \mathbb R \qquad \text{ with }(x,y) \mapsto x+y$$
That is, the plus function has two real numbers as input, and one real number as output.

The relation "=" defined on the real numbers is the set:
$$\{ (x,x) : x \in \mathbb R \}$$
That is, two real numbers have an equality relation with each other if they are the same number.
In your case "A+1" has an equality relation with "B+C".

Btw, "=" can also be considered an operation.
In that case both "A+1" and "B+C" are inputs, and the output is either "true" or "false".

3. Jan 5, 2013

### benno

Thanks very much for the clear explanation. That was exactly what I was looking for.