What defines an operator input/output for simple expressions

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In the discussion, the operator "+" acts on inputs "B" and "C," with the output being the expression "B+C." The equals sign "=" is identified as a relation that compares the left-hand side (LHS) and right-hand side (RHS) expressions, and it can also be considered an operator with both inputs producing a true or false output. The mathematical function for addition is defined as taking two real numbers and producing a single real number. The equality relation indicates that two numbers are equal if they are the same. Overall, the conversation clarifies the roles of operators and relations in mathematical expressions.
benno
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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?
 
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Welcome to PF, benno! :smile: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".
 
Thanks very much for the clear explanation. That was exactly what I was looking for.
 
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