Relation
High school
A relation is a rule that determines whether whether two objects are related.
Undergraduate
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
Worked examples
-
The relation $<$ on $\reals$
: $\setdef{(x, y) \in \reals \times \reals \suchthat x \text{ is less than } y}$
Undergraduate
The relation $<$ on $\reals$ is the set of pairs $(x, y)$ such that $x$ is less than $y$, so $2 < 3$ is true because $(2, 3)$ is in the set, and $3 < 2$ is false because $(3, 2)$ is not in the set.
Examples
-
$=$, $<$, and $\subseteq$
High school
$=$, $<$, and $\subseteq$ are relations because they make statements that are true or false: “$2 < 3$” is true and “$4 < 4$” is false.
-
The relation $<$ on $\reals$
Undergraduate
The relation $<$ on $\reals$ is the set of pairs $(x, y)$ such that $x$ is less than $y$, so $2 < 3$ is true because $(2, 3)$ is in the set, and $3 < 2$ is false because $(3, 2)$ is not in the set.
Non-examples
Things that may be mistaken for examples.
-
Addition $+$ on $\reals$
Elementary
Addition on $\reals$ is not a relation, because “$2 + 4$” does not have a true or false value.
-
Multiplication $\times$ on $\reals$
Elementary
Multiplication on $\reals$ is not a relation, because “$2 \times 4$” does not have a true or false value.
- “$7$ is prime” High school
-
Subtraction on $\nats$
High school
Subtraction is not a relation, because “$5 - 3$” is not true or false.
-
$\cup$ on the subsets of a set $X$
High school
$\cup$ is not a relation, because “$A \cup B$” is not true or false.
-
“$x > 3$”
High school
“$x > 3$” is not a relation, which is a set of pairs.
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).