Question

Show that the relation "less than or equal to" on the set of integers Z, is reflexive and transitive
but not symmetric
​

Answer 1

Basic Concept :-

Reflexive :-

• Relation is reflexive. If (a, a) ∈ R for every a ∈ A.

Symmetric :-

• Relation is symmetric, If (a, b) ∈ R, then (b, a) ∈ R.

Transitive :-

• Relation is transitive, If (a, b) ∈ R & (b, c) ∈ R, then (a, c) ∈ R. If relation is reflexive, symmetric and transitive

Let's solve the problem now!!

Here,

Relation, R mathematically represented as

$\rm :\longmapsto\:R \: = \{(a, b) : a \leqslant b \: where \: a, b \: \in \: Z \}$

Reflexive :-

$\rm :\longmapsto\:Let \: a \: \in \: Z$

$\rm :\longmapsto\:We\:know,\: a = a$

$\rm :\longmapsto\:(a, a) \in \: R$

$\bf\implies \:R \: is \: reflexive$

Symmetric :-

$\rm :\longmapsto\:Let \: a, b \in \: Z \: such \: that \: (a, b) \in \: R$

$\rm :\implies\:a \leqslant b$

$\rm : \cancel {\implies} \: \: b \leqslant a$

$\rm :\implies\:(b, a) \: \cancel\in \: R$

$\bf\implies \:R \: is \: not \: symmetric$

Transitive :-

$\rm :\longmapsto\:Let \: a, b, c \in \: Z \: such \: that(a, b) \in \: R \: and \: (b, c) \in \: R$

$\rm :\longmapsto\:as \: (a, b) \in \: R\rm :\implies\:a \leqslant b - - (1)$

$\rm :\longmapsto\:as \: (b, c) \in \: R \rm \: \implies\:b \leqslant c - - (2)$

From equation (1) and equation (2), we concluded that

$\rm :\longmapsto\:a \leqslant c$

$\rm :\implies\:(a, c) \in \: R$

$\bf\implies \:R \: is \: trasitive.$