Domain, range and inverse of a relation

A rela­tion from $A$ to $B$ need not use every ele­ment of $A$ or every ele­ment of $B$. The ele­ments that actu­ally take part form the domain and the range of the rela­tion. Revers­ing every pair of a rela­tion gives another rela­tion, called its inverse. These three ideas carry straight over to func­tions, where they are used all the time.

Domain and range#

Let $R$ be a rela­tion from $A$ to $B$. Then:

1. The domain of $R$ is the set of first entries of the pairs in $R$: $\{a \in A : aRb \text{ for some } b \in B\}$.
2. The range of $R$ is the set of sec­ond entries of the pairs in $R$: $\{b \in B : aRb \text{ for some } a \in A\}$.

The whole set $B$ is some­times called the codomain of $R$. The range is always a sub­set of the codomain, and the domain is a sub­set of $A$.

Exam­ple 1#

Let $A = \{1, 2, 3\}$ and $B = \{a, b\}$, so that $A \times B = \{(1, a), (1, b), (2, a), (2, b), (3, a), (3, b)\}$. Let $R_1 = \{(1, a), (2, a)\}$. Find the domain and range of $R_1$.

The first entries are 1 and 2, so the domain of $R_1$ is $\{1, 2\}$. The only sec­ond entry is $a$, so the range of $R_1$ is $\{a\}$. The ele­ment 3 of $A$ and the ele­ment $b$ of $B$ take no part.

Exam­ple 2#

Let $A = \{1, 2, 3, 4, 5, 6\}$ and let $R$ be the rela­tion on $A$ given by $R = \{(a, b) : b = a + 2\}$. Write $R$ as a set of pairs, and find its domain and range.

For $a = 1, 2, 3, 4$ the num­ber $a + 2$ is in $A$; for $a = 5, 6$ it is not. So

$$R = \{(1, 3), (2, 4), (3, 5), (4, 6)\}$$

The domain is $\{1, 2, 3, 4\}$ and the range is $\{3, 4, 5, 6\}$.

Exam­ple 3#

Let $R = \{(x, y) : x, y \in N, \ x + 2y = 10\}$. Find $R$, its domain and its range.

Since $x = 10 - 2y$ must be a nat­ural num­ber, $y$ can be 1, 2, 3 or 4 (for $y = 5$, $x = 0$, which is not nat­ural). This gives

$$R = \{(8, 1), (6, 2), (4, 3), (2, 4)\}$$

The domain is $\{2, 4, 6, 8\}$ and the range is $\{1, 2, 3, 4\}$.

Arrow dia­grams#

A rela­tion between small sets can be shown by an arrow dia­gram: draw two ovals, list the ele­ments of $A$ in the first and those of $B$ in the sec­ond, and draw an arrow from $a$ to $b$ for each pair $(a, b)$ in the rela­tion. The domain is the set of ele­ments in the first oval that have an arrow leav­ing them; the range is the set of ele­ments in the sec­ond oval that have an arrow arriv­ing.

The inverse of a rela­tion#

If $R$ is a rela­tion from $A$ to $B$, then the inverse rela­tion $R^{-1}$ is the rela­tion from $B$ to $A$ obtained by revers­ing every pair:

$$R^{-1} = \{(b, a) : (a, b) \in R\}$$

In an arrow dia­gram, the inverse sim­ply reverses every arrow.

Exam­ple 4#

Let $A = \{1, 2, 3\}$, $B = \{a, b\}$ and $R_1 = \{(1, b), (3, b)\}$. Find $R_1^{-1}$.

Reverse each pair: $R_1^{-1} = \{(b, 1), (b, 3)\}$. It is a rela­tion from $B$ to $A$.

Since revers­ing swaps first and sec­ond entries:

• the domain of $R^{-1}$ is the range of $R$, and
• the range of $R^{-1}$ is the domain of $R$.

Revers­ing twice gives back the orig­i­nal rela­tion: $(R^{-1})^{-1} = R$.

Exam­ple 5#

For $R = \{(1, 3), (2, 4), (3, 5), (4, 6)\}$ from Exam­ple 2, find $R^{-1}$ with its domain and range, and describe it by a rule.

$R^{-1} = \{(3, 1), (4, 2), (5, 3), (6, 4)\}$. Its domain is $\{3, 4, 5, 6\}$, the range of $R$; its range is $\{1, 2, 3, 4\}$, the domain of $R$. Since $R$ was "$b = a + 2$", the inverse is "$b = a - 2$": each ele­ment is related to the num­ber 2 less than it.

Exam­ple 6#

Let $R = \{(x, x^2) : x \text{ is a prime number less than } 10\}$. Find $R$, $R^{-1}$, and the domain and range of each.

The primes less than 10 are 2, 3, 5, 7. So $R = \{(2, 4), (3, 9), (5, 25), (7, 49)\}$, with domain $\{2, 3, 5, 7\}$ and range $\{4, 9, 25, 49\}$.

$R^{-1} = \{(4, 2), (9, 3), (25, 5), (49, 7)\}$, with domain $\{4, 9, 25, 49\}$ and range $\{2, 3, 5, 7\}$.

Prac­tice#

1. Find the domain and range of $R = \{(1, 2), (1, 3), (2, 5), (4, 5)\}$.
2. Let $A = \{1, 2, 3, 4, 5\}$ and $R = \{(a, b) : a, b \in A, \ b = 2a - 1\}$. Write $R$ and find its domain and range.
3. Let $R = \{(x, y) : x, y \in N, \ 2x + y = 9\}$. Write $R$ and find its domain and range.
4. Find the inverse of $R = \{(a, 1), (b, 2), (c, 2)\}$.
5. If the domain of a rela­tion $R$ is $\{1, 2, 3\}$ and its range is $\{7, 8\}$, what are the domain and range of $R^{-1}$?
6. Let $A = \{1, 2, 3, 4, 6\}$ and $R = \{(a, b) : a, b \in A, \ a \text{ divides } b, \ a \neq b\}$. Write $R$ and find its domain and range.
7. Let $R = \{(x, y) : x \in \{1, 2, 3\}, \ y = x^2 + 1\}$. Write $R$ and $R^{-1}$.
8. The rela­tion $R$ on $\{1, 2, 3, 4\}$ is "$a < b$". Write $R$ and $R^{-1}$, and describe $R^{-1}$ in words.

Answers#