Expression

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In mathematics, an expression is any meaningful combination of symbols. What this means exactly varies depending on the mathematical context. For instance, arithmetic expressions typically consist of numbers, variables, and operators, arranged in a sensible way. Thus, $3 - \frac x4$ is an arithmetic expression, while $7 \times + 43$ is not. There are no equal signs in expressions.

Again depending on context, one is often interested in finding equivalences between expressions of various sorts. In standard arithmetic, for instance, the two expressions $(3x + 4) - x + 2$ is equivalent to the expression $2x + 6$. This is represented by the use of an equal sign, $3x + 4 -x +2 = 2x - 6$. In other branches of mathematics, other symbols are sometimes used, especially the symbol $\equiv$.

Note that in arithmetic, an equality like the one above is not an expression. In mathematical logic, however, arithmetic equations often are expressions. For instance, $3x + 2 = 11$ is a valid expression in Peano arithmetic (with a proper interpretation of symbols), and is logically equivalent to the expression $x = 3$. We might write this equivalence of expressions as $\left(3x +2 = 11\right) \Longleftrightarrow \left(x = 3\right)$. Just like the equality of two arithmetic expressions is not an arithmetic expression, this equivalence of logical expressions is not a logical expression.

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