The definition of a relation in mathematics, along with its domain and range, is presented with examples, questions and their solutions.

A relation is a correspondence between two sets called the domain and the range [1].

Example 1

The relation \( R_1 \) between a group of four students \( \{ \; \text{Sasha}\; , \; \text{Smith} \; , \; \text{Jane} \; , \; \text{John} \; \} \) and their exam scores \( \{ \; 85 \; , \; 92 \; , \; 71 \; \} \) is shown in the diagram below.

The above relation \( R_1 \) may be represented as a set of ordered pairs as follows: \( \quad R_1 = \{ \; (\text{Sasha} \; , \; 85) \; , \; (\text{Smith} \; , \; 85) \; , \; (\text{Jane} \; , \; 92) \; , \; (\text{John} \; , \; 71) \; \} \)

The first element in each ordered pair is an element of the domain and the second element in the pair is the corresponding element of the range.

The domain \( D \) of the relation \( R_1 \) is a set given by: \( \quad D = \{ \; \text{Sasha}\; , \; \text{Smith} \; , \; \text{Jane} \; , \; \text{John} \; \} \)

The range \( R \) of the relation \( R_1 \) is a set given by: \( \quad R = \{ \; 85 \; , \; 92 \; , \; 71 \; \} \)

Example 2

The relation \( R_2 \) between the time at which matches are to be played \( \{ 10 \; \text{am} \; , \; 1 \; \text{pm} \; , \; 4 \; \text{pm} \} \) and the pairs of teams \( \{ \text{A vs B} \; , \; \text{C vs D} \; , \; \text{E vs F} \; , \; \text{G vs H} \} \) to play at that time is shown below.

The above relation \( R_2 \) may be represented as a set of ordered pairs as follows: \( \quad R_2 = \{ \; (10 \; \text{am} \; , \; \text{A vs B}) \; , \; (1 \; \text{pm} \; , \; \text{C vs D}) \; , \; (4 \; \text{pm} \; , \; \text{E vs F}) \; , \; (4 \; \text{pm} \; , \; \text{G vs H}) \; \} \)

Note that the first element in the ordered pair is an element of the domain and the second element in the pair is the corresponding element of the range.

The domain \( D \) of the relation \( R_2 \) is a set given by: \( \quad D = \{ \; 10 \; \text{am} \; , \; 1 \; \text{pm} \; , \; 4 \; \text{pm} \; \} \)

The range \( R \) of the relation \( R_2 \) is a set given by: \( \quad R = \{ \; \text{A vs B} \; , \; \text{C vs D} \; , \; \text{E vs F} \; , \; \text{G vs H} \; \} \)

Using the same relation, we describe some of the most important representations of relations in mathematics.

Relations use a set for the domain and a set for the range as shown in figure 3 and arrows from each element in \( D \) to the corresponding element in the range \( R \).

A relation may also be represented as a set of ordered pairs as shown below. The first element in an ordered pair is an element in the domain \( D \) and the second element is the corresponding element in the range \( R \). \( R_3 = \{ \; (2,3) \; , \; (4,1) \; , \; (6,4) \; , \; (7,2) \; , \; (7,6) \; \} \)

A relation may be represented by a table with two rows (or two columns). A row \( x \) with elements in the domain \( D \) and a row \( y \) with the corresponding elements in the range \( R \) as shown below.

\( x \) | \( 2 \) | \( 4 \) | \( 6\) | \( 7 \) | \( 7 \) |

\( y \) | \( 3 \) | \( 1 \) | \( 4 \) | \( 2 \) | \(6 \) |

Use a system of rectangular coordinates where each ordered pair is represented by a point \( (x,y) \) where \( x \) is an element in the domain \( D \) and \( y \) is the corresponding element in the range \( R \).

The equation \( y = 3 x + 1 \) represents a relation between \( x \) and \( y \). Given values of the variable \( x \) in the domain \( D \) of the relation, we can find the corresponding values of \( y \) in the range \( R \).

For example, for \( x = \color{red}{2} \), we calculate the corresponding value of \( y \) by substituting \( x \) by \( \color{red}{2} \) in the given equation as follows:

\( y = 3(\color{red}{2}) + 1 \)

We simplify to obtain:

\( y = 7 \)

If we are given several values of \( x \), it is better to use a table as follows:

\( x \) | \( \color{red}{-3} \) | \( \color{red} 0 \) | \( \color{red} 5\) | \( \color{red} 8 \) |

\( y \) | \( 3(\color{red}{-3}) + 1 = - 8 \) | \( 3(\color{red}{0}) + 1 = 1\) | \( 3(\color{red}{5}) + 1 = 16\) | \( 3(\color{red}{8}) + 1 = 25\) |

Example 3

Given the relation \( R_4= \{ \; (-1 , 3) \; , \; (2 , 4) \; , \; (5,7) \; , \; (2 , 6) \; \} \) as a set of ordered pairs,

a) find its domain and range,

and represent it as

b) a graph

c) table

d) a Venn diagram.

__Solution to Example 3__

a)

The relation \( R_4 \) is given as a set of ordered pairs: \( \{ \; (\color{red}{-1} , \color{blue}{3}) \; , \; (\color{red}{2} , \color{blue}{4}) \; , \; (\color{red}{5},\color{blue}{7}) \; , \; (\color{red}{2} , \color{blue}{6}) \; \} \)

The domain \( D \) is the set of all values of the first element in the ordered pairs.

\( \color{red}{D = \{ \; -1 , 2 , 5 \; \}} \)

The domain \( R \) is the set of all values of the second element in the ordered pairs.

\( \color{blue}{R = \{ \; 3 , 4 , 6 , 7 \; \}} \)

b)

Relation \( R_4 \) represented by a graph is shown in figure 5 below where each point correspond to an ordered pair in the given relation.

c)

The given relation \( R_4 \) represented by a table is shown below

\( x \) | \( -1 \) | \( 2 \) | \( 2 \) | \( 5 \) |

\( y \) | \( 3 \) | \( 4 \) | \( 6 \) | \( 7 \) |

d)

Relation \( R_4 \) represented by a Venn diagram is shown in figure 5 below.

Relation \( R_5 \) is given by its graph in figure 6 below.

a) Represent relation \( R_5 \) a) a set of ordered pairs, b) as a table, b) , c) using Venn diagram.

b) Find its domain and range.

Relation \( R_6 \) is given by a table below.

\( x \) | \( -2 \) | \( -2 \) | \( -2 \) | \( 0 \) | \( 1 \) | \( 3 \) |

\( y \) | \( 3 \) | \( 4 \) | \( 5 \) | \( 4 \) | \( 0 \) | \( 0 \) |

a) Find its domain and range.

Represent it by b) a set of ordered pairs, c) a Venn diagram.

Given answers in ordered pairs representation.

a) Give an example of two different relations with the same domain.

b) Give an example of two different relations with the same range.

c) Give an example of two different relations with the same domain and range.

A relation is defined by the equation \( y + 4 x = 2 \) where \( x \) is a variable in the domain of the relation taking the values: \( \{ -1 , 0 , 1 , 2 \} \).

a) Use a table to find the \( y \) values corresponding to the given \( x \) values.

b) Represent the relation using a graph.

Relation \( R_5 \) as a set of ordered pairs is given by: \( \quad R_5 = \{ \; (-2,2) \; , \; (1,1) \; , \; (2,1) \; , \; (2,5) \; , \; (4,2) \; , \; (4,3) \; , \;(4,4) \; \} \)

b)

Relation \( R_5 \) as table

\( x \) | \( -2 \) | \( 1 \) | \( 2 \) | \( 2 \) | \( 4 \) | \( 4 \) | \( 4 \) |

\( y \) | \( 2 \) | \( 1 \) | \( 1 \) | \( 5 \) | \( 2 \) | \( 3 \) | \( 4 \) |

Relation \( R_5 \) using Venn diagram

a) The domain \( D \) is the set of all first terms in the ordered pair: \( D = \{ -2 \; , \; 0 \; , \; 1 \; , \; 3 \} \)

The range \( R \) is the set of all second terms in the ordered pair: \( R = \{ 3 \; , \; 4 \; , \; 5 \; , \; 0 \} \)

b) Relation \( R_6 \) as an ordered pair: \( \{ \; (-2,3) \; , \; (-2,4) \; , \; (-2,5) \; , \; (0,4) \; , \; (1,0) \; , \; (3,0) \; \}\)

c) Relation \( R_6 \) represented by Venn diagram

a) The relations \( \; (-3,3) \; , \; (-2,4) \; , \; (0,5) \; , \; (4,5) \; \) and \( \; (-3,3) \; , \; (-2,8) \; , \; (0,5) \; , \; (4,0) \; \) have the same domain but are different.

b) The relations \( \; (0,3) \; , \; (-2,5) \; , \; (6,5) \; , \; (7,5) \; \) and \( \; (-3,3) \; , \; (-2,5) \; , \; (0,5) \; , \; (9,5) \; \) have the same range but are different.

c) The relations \( \; (-1,0) \; , \; (0,-5) \; , \; (4,5) \; , \; (7,8) \; \) and \( \; (-1,0) \; , \; (-1,-5) \; , \; (0,-5) \; , \; (4,5) \; , \; (7,8) \; \) have the same domain and range but are different.

The relation is defined by the equation: \( y + 4 x = 2 \) which may be written as \( y = - 4x + 2 \)

a)
The variable \( x \) takes the values: \( \{ -1 , 0 , 1 , 2 \} \), hence the table of values of \( x \) and corresponding \( y \) calculated using the given equation as follows:

\( x \) | \( -1 \) | \( 0 \) | \( 1 \) | \( 2 \) |

\( y \) | \( y = - 4(-1) + 2 = 6 \) | \( y = - 4(0) + 2 = 2 \) | \( y = - 4(1) + 2 = -2 \) | \( y = - 4(2) + 2 = -6 \) |

b) The graph corresponding to the relation defined by the equation \( y + 4 x = 2 \) and with domain \( \{ -1 , 0 , 1 , 2 \} \) is shown below.

Find Domain and Range of Relations Given by Graphs

Find The Inverse of a Relation