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).
Definitions
A binary operation is a way of combining two numbers to get another number.
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
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Addition $+$ on $\reals$
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 $\reals$
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 $X$
$\cup$ is a binary operation on the subsets of $X$ because the union of two subsets of $X$ is a subset of $X$.
Non-examples
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$=$, $<$, and $\subseteq$
$=$, $<$, and $\subseteq$ are not binary operations, because they do not produce a new number or set.
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Subtraction on $\nats$
Subtraction is not a binary operation on $\nats$, because $3 - 5$ is not in $\nats$ (it is one on $\mathbb{Z}$).
A relation is a rule that determines whether whether two objects are related.
Consider sets $X$ and $Y$ and their product $X \times Y$. A binary relation between $X$ and $Y$ is a subset $R \subseteq X \times Y$. For $x \in X$ and $y \in Y$, the statement $x \mathrel{R} y$ (“$x$ is related to $y$”) is true if $(x, y) \in R$ and false if $(x, y) \notin R$.
Examples
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$=$, $<$, and $\subseteq$
$=$, $<$, and $\subseteq$ are relations because they make statements that are true or false: “$2 < 3$” is true and “$4 < 4$” is false.
Non-examples
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Addition $+$ on $\reals$
Addition on $\reals$ is not a relation, because “$2 + 4$” does not have a true or false value.
-
Multiplication $\times$ on $\reals$
Multiplication on $\reals$ is not a relation, because “$2 \times 4$” does not have a true or false value.
- “$7$ is prime”
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Subtraction on $\nats$
Subtraction is not a relation, because “$5 - 3$” is not true or false.
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$\cup$ on the subsets of a set $X$
$\cup$ is not a relation, because “$A \cup B$” is not true or false.
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“$x > 3$”
“$x > 3$” is not a relation, which is a set of pairs.
Examples
| Example | binary operation | relation | Why |
|---|---|---|---|
| $=$, $<$, and $\subseteq$ | ✗ | ✓ | |
| Addition $+$ on $\reals$ | ✓ | ✗ | |
| Multiplication $\times$ on $\reals$ | ✓ | ✗ | |
| $\cup$ on the subsets of a set $X$ | ✓ | ✗ | |
| Subtraction on $\nats$ | ✗ | ✗ |
Notes
- “$2 + 3 = 5$” combines both: $+$ is an operation that produces $5$, and $=$ is a relation that makes the whole statement a proposition.
- “Operator” is often used for “operation” in elementary arithmetic (the “plus operator”), but in analysis an operator is a map between vector spaces. See types of maps.