Binary operation
Elementary
A binary operation is a way of combining two numbers to get another number.
Undergraduate
A binary operation on a set $S$ is a function $\ast : S \times S \to S$. The output must be in the same set $S$ (the set is “closed” under the operation).
Examples
Examples
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Addition $+$ on $\reals$
Elementary
Addition on $\reals$ is a binary operation, because addition combines two real numbers into a new one: “$2 + 4$” produces $6$.
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Multiplication $\times$ on $\reals$
Elementary
Multiplication on $\reals$ is a binary operation, because multiplication combines two real numbers into a new one: “$2 \times 4$” produces $8$.
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$\cup$ on the subsets of a set $X$
High school
$\cup$ is a binary operation on the subsets of $X$ because the union of two subsets of $X$ is a subset of $X$.
Non-examples
Things that may be mistaken for examples.
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$=$, $<$, and $\subseteq$
High school
$=$, $<$, and $\subseteq$ are not binary operations, because they do not produce a new number or set.
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Subtraction on $\nats$
High school
Subtraction is not a binary operation on $\nats$, because $3 - 5$ is not in $\nats$ (it is one on $\mathbb{Z}$).
- The relation $<$ on $\reals$ Undergraduate
Distinctions involving this term
- Operation vs. relation: Symbols placed between two objects, such as $+$ and $<$, do different kinds of things. A binary operation takes two objects and produces a new object of the same type. A binary relation takes two objects and says whether they are related (true or false).