Relation | Lesson Note

Relation | Grade 9 Algebra | NMA

Part I — Discovering Relations

Activity

Let us consider two non-empty sets:

A = \{1, 2, 3\} \qquad B = \{1, 2\}

Find the Cartesian Product \( A \times B \):

A \times B = \{1,2,3\} \times \{1,2\} = \{(1,1),\,(1,2),\,(2,1),\,(2,2),\,(3,1),\,(3,2)\}

Selection – Form sub-collections:

1. Collect ordered pairs where \(x = y\), name it \(R_1\):
\( R_1 = \{(1,1),\;(2,2)\} \)
2. Collect ordered pairs where \(x > y\), name it \(R_2\):
\( R_2 = \{(2,1),\;(3,1),\;(3,2)\} \)
3. Collect ordered pairs where \(x < y\), name it \(R_3\):
\( R_3 = \{(1,2)\} \)

Observation

Looking at our sub-collections, we can see that:

\(R_1,\; R_2,\; R_3\) are collections of ordered pairs.
\(R_1 \subset A \times B\)
\(R_2 \subset A \times B\)
\(R_3 \subset A \times B\)

Conclusion

Such a collection of ordered pairs \((x, y)\) where \(x\) and \(y\) have some connection between them, and which is also a subset of \(A \times B\), is commonly known as a Relation.

Definition of Relation

Definition

Let \(A\) and \(B\) be any two non-empty sets. A relation \(R\) defined from set \(A\) to \(B\) is a collection of ordered pairs \((x, y)\) from \(A \times B\) where \(x\) and \(y\) have a meaningful relation.

R = \{(x,\,y) : x \in A,\; y \in B\} \subset A \times B

Common Rules used in relations: "Is greater than", "Is less than", "Is equal to", "Is square of", etc.

Domain, Range, and Co-domain

Domain of R

The set of all first elements (antecedents) from the ordered pairs that belong to the relation.

Range of R

The set of all second elements (consequents) from the ordered pairs that belong to the relation.

Co-domain of R

If \(R\) is a relation defined from set A to B and \(R \subset A\times B\), then set B is called the co-domain of relation \(R\).

Example: If \( R = \{(1,1),\;(2,2),\;(3,3)\} \)

Domain of R \(= \{1,\;2,\;3\}\)
Range of R \(= \{1,\;2,\;3\}\)

Inverse Relation

Definition

If \(R\) is a relation from set \(A\) to set \(B\), then the inverse relation of \(R\), denoted by \(R^{-1}\), is obtained by reversing the ordered pairs.

If \((a,b) \in R\) then \((b,a) \in R^{-1}\)

Mathematical Form:

\text{If } R = \{(a,b) \mid a \in A,\ b \in B\} \] \[ \text{then } R^{-1} = \{(b,a) \mid b \in B,\ a \in A\}

Example

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

Then \( R^{-1} = \{(2,1),\ (3,2),\ (5,4)\} \)

Ways of Representation of a Relation

Setup: Let \( A = \{1,2,3\} \) and \( B = \{1,2\} \).
\( A \times B = \{(1,1),(1,2),(2,1),(2,2),(3,1),(3,2)\} \).
Define relation \( R \) as "Is equal to".

R = \{(1,\;1),\;(2,\;2)\}

\( x \)	1	2
\( y \)	1	2

R = \{(x,\,y) : x = y,\; x \in A,\; y \in B\}

Types of Relations

Context: Let \( A = \{1, 2, 3\} \). Then:
\( A \times A = \{(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)\} \)

Reflexive Relation

A relation which contains ordered pairs \((x,y)\) such that for all \((x,y) \in R\), either \((x,x) \in R\) or \((y,y) \in R\).

\( R = \{(1,1),\;(2,2),\;(3,3)\} \)
Every element relates to itself.

Symmetric Relation

A relation which contains ordered pairs \((x,y)\) such that for all \((x,y) \in R\), \((y,x) \in R\).

\( R = \{(1,2),(2,1),(1,3),(3,1),(2,3),(3,2)\} \)
If a relates to b, then b relates to a.

Transitive Relation

A relation where for all \((x,y) \in R\) and \((y,z) \in R\), we have \((x,z) \in R\).

\( R = \{(1,2),\;(2,3),\;(1,3)\} \)
If a → b and b → c, then a → c.

Equivalence Relation

A relation \(R\) is said to be an Equivalence Relation if and only if it is simultaneously:

Reflexive
Symmetric
Transitive

Example: \( R = \{(1,1),\;(1,2),\;(2,1),\;(2,3),\;(1,3)\} \)