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Is everything that you can write in math (that makes mathematical sense) an expression? If not, what would be examples of non-expressions? And would all expressions be composed of expressions themselves?

Also, are operators (like the differential operator) by themselves expressions?

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What do you mean by "expression"? – Harry Stern Jan 16 '11 at 19:03
@Harry whatever mathematicians like to use it for. – wrongusername Jan 16 '11 at 19:06
@wrongusername: the term does not, to my knowledge, have a fixed meaning in mathematics. – Qiaochu Yuan Jan 16 '11 at 19:20
Look up for an example of defining what an "expression" is in one part of mathematics. – Yuval Filmus Jan 16 '11 at 20:04

According to Merriam-Webster's Collegiate Dictionary, in mathematics, an expression is "a mathematical or logical symbol or a meaningful combination of symbols". Thus $a$, $+$, $a+b$, and $a+b=c$ are expressions, while $+\!=$ and $==$ are not. Whether an arbitrary fragment of a meaningful expression, such as $=c$, is necessarily still an expression is dubious. I am inclined to think not, but would agree with the statement that all expressions are themselves composed of expressions, down to the (atomic) level of single symbols.

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No, operators are not expressions. For me, an expression is a statement (like a sentence is a statement). It's a thing you can assign a truth value to. If you can't assign a truth value to it (i.e. say it's true or false) then it's not an expression.

Operators, logic symbols, etc. act like conjunctions and punctuation marks which make sentences easier to read and frame context.

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I think that's more usual to consider that an expression is more general than a statement; the later is what has a truth value. But an expression can be the RHS of an equality. – leonbloy Feb 5 '14 at 1:03

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