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Mathematics LibreTexts

2: Binary Relations

Definition

Let \(S\) be a non-empty set. A binary relation \(R\) on \(S\) is a rule that relates two elements in \(S\).

 In terms of set,  \( R \subset S\times S.\)

Example \(\PageIndex{1}\):

Let \(S=\{1,2,3\}\), and let \(R\) be the relation \( <\).

Then  

  1. \(1\) is related to \(2\)., we denote this by \( 1 R 2 \) or \(1 < 2\).
  2. \(2\) is not related to \(1\)., we denote this by \( 2 \not{R} 1 \) or \(2 \nless \).1

Example \(\PageIndex{2}\): 

The following are binary relations on \mathbb Z.

  1. \(=\)
  2. \(<\)
  3. \(>\)
  4. \(\leq\)
  5. \(geq\)

 

 

 

Definition

Let \(a\) and \(d\) be integers. The \(d\) divides \(a\), denoted by \(d|a\), if there exists an integer \(k\) such that \(a=dk.\) 

In other words,

  •  \(a\) is multiples of \(d\)
  • \(d\) is a factor of \(a\)
  • \(a\) is divisible by \(d\)

Example \(\PageIndex{3}\): 

State whether the following is true or false. Justify your answer.

  1. \( 4|2\)
  2. \(2|4\)

Example \(\PageIndex{4}\):

Define  a relation \(R\) on \(S={\mathbb Z}\) by:
\(a \,R\, b\) if and only if \( 2 \mid a-b.\)

Properties

Definition: Reflexive

Let \(R\) be relation on a set \(S\), then \(R\) is said to be reflexive if  \(a R a, \forall a \in S\).

Example \(\PageIndex{6}\):

Define  a relation \(R\) on \(S={\mathbb Z}\) by:
\(a \,R\, b\) if and only if \( 2 \mid a-b.\)

Is \(R\) Reflexive?