Name: Definition of a Set
Uploaded: 2019-07-19
Duration: 17 min 33 s
Description: A set is a collection of distinct elements with no specific order, where each element is unique and well-defined. Watch this video to understand.

Question

Prove that the relation &ldquo;less than&rdquo; in the set of natural number is transitive but not reflexive and symmetric.

Accepted Answer

We have to prove that the relation “less than” in the set of natural number is transitive but not reflexive and symmetric.

Checking Reflexive relation:

A relation $R$ , on a set $A$ , is said to be reflexive if every element of $A$ , is related to itself. Thus, $R$ , on a set $A$ , is not reflexive if there exists an element $a \in A$ , such that $(a, a) \notin R$

Let $a \in N$ , if $a < a$ then $(a, a) \notin R$ , (Since $a = a$ , is not possible).

So, the relation can not be reflexive.

Checking Symmetric relation :

A relation $R$ , on a set $A$ , is said to be a symmetric relation
if $(a, b) \in R \Rightarrow (b, a) \in R$ , for all $a, b \in A .$ i.e. $a R b \Rightarrow b R a$ , for all $a, b \in A$

Let $a \in N$ and $b \in N$ , so if $a < b$

then $b ≮ a$ , $\Rightarrow (a, b) \in R$ but $(b, a) \notin R$ , so given relation is not symmetric.

Checking Transitive relation: Let $A$ , be any set. A relation $R$ on $A$ , is said to be a transitive relation if $(a, b) \in R$ and $(b, c) \in R \Rightarrow (a, c) \in R$ , for all $a, b, c \in A$ i.e. $a R b$ and $b R c \Rightarrow a R c$ , for all $a, b, c \in A$ .

Let $a < b$ and $b < c$ $\Rightarrow a < c$ , ( $∵ a, b, c \in N$ )

then $(a, b) \in R$ and $(b, c) \in R$ $\Rightarrow (a, c) \in R$ .

$∴$ Given relation, $R$ , is not reflexive and symmetric but transitive.

Prove that the relation “less than” in the set of natural number is transitive but not reflexive and symmetric.

Important Questions on Set Theory and Relations

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