Let A ={1,2,3} . number of relations on A , containing (1,2) and (2,3) , which are reflexive and tra

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1.Relation and Function

medium

Let $A =\{1,2,3\}$. The number of relations on $A$ , containing $(1,2)$ and $(2,3)$, which are reflexive and transitive but not symmetric, is _____

A$3$

B$4$

C$5$

D$6$

(JEE MAIN-2025)

Solution

Transitivity
$(1,2) \in R,(2,3) \in R \Rightarrow(1,3) \in R$
For reflexive $(1,1),(2,2)(3,3) \in R$
Now $(2,1),(3,2),(3,1)$
$(3,1)$ cannot be taken
$(1) (2,1)$ taken and $(3,2)$ not taken
$(2) (3,2)$ taken and $(2,1)$ not taken
$(3)$ Both not taken
therefore $3$ relations are possible.

Standard 12

Mathematics

Let $A =\{2,3,4,5, \ldots ., 30\}$ and $^{\prime} \simeq ^{\prime}$ be an equivalence relation on $A \times A ,$ defined by $(a, b) \simeq (c, d),$ if and only if $a d=b c .$ Then the number of ordered pairs which satisfy this equivalence relation with ordered pair $(4,3)$ is equal to :

hard

(JEE MAIN-2021)

Let $P = \{ (x,\,y)|{x^2} + {y^2} = 1,\,x,\,y \in R\} $. Then $P$ is

easy

How many reflexive relation are there on a set ' with $3$ elements

normal

Determine whether each of the following relations are reflexive, symmetric and transitive:

Relation $\mathrm{R}$ in the set $\mathrm{N}$ of natural numbers defined as

$\mathrm{R}=\{(x, y): y=x+5 $ and $ x<4\}$

medium

Let $S=\{1,2,3, \ldots, 10\}$. Suppose $M$ is the set of all the subsets of $S$, then the relation $R=\{(A, B): A \cap B \neq \phi ; A, B \in M\}$ is :

hard

(JEE MAIN-2024)