Question

The relation $\le$ on numbers has the following properties:
(i). $a\le a\forall a\in R$ (Reflexivity)
(ii). If $a\le b$ and $b\le a$ then $a=b\forall a,b\in R$ (Anti-symmetry)
(iii). If $a\le b$ and $b\le c$ then $a\le c\forall a,b\in R$ (Transitivity)
Which of the above properties the relation $\subset$ on $p(A)$ has?
1). $(i)$ and $(ii)$
2). $(i)$ and $(iii)$
3). $(ii)$ and $(iii)$
4). $(i)$ , $(ii)$ and $(iii)$

Answer 1

To solve this problem, we have to understand the concept of relation and its types and after that we will check the given relation for all reflexivity, anti-symmetry and transitivity and then after checking these all conditions, we will get our required answer.

Complete step-by-step solution:
Relation can be defined as a connection between the elements of two or more sets, and all the sets must be non-empty. Relations can be represented in three ways: Roaster form, Set-builder Form, by arrow diagram.
Types of relations are as: Empty Relation, Reflexive Relation, Transitive Relation, Anti-symmetric Relation, Universal Relation, Inverse Relation, and Equivalence Relation.
Let’s understand the reflexive relation, anti-symmetric relation and Transitive relation.
A relation is said to be reflexive if every element of set $M$ maps to itself only (i.e. for every $p\in M$ , $(p,p)\in R$ , here $R$ represents the relation set.
A relation is said to be anti-symmetric if $(a,b)\in R$ and $(b,a)\in R$, that means $a=b$ . And if it does not follow above, it is not antisymmetric.
A relation $R$ is said to be transitive if $(a,b)\in R,(b,c)\in R$ , then $(a,c)\in R$ such that for all $a,b,c\in A$ , here $A$ is set of all elements.
Now according to the given question, we have to check reflexivity, anti-symmetry and transitivity of the relation $\subset$ .
For reflexivity:
For relation $\subset$ ,
We can say that $x\subset x$ (because we know every set is a subset of itself)
So, we can say that for this relation Reflexivity is true.
For Anti-symmetry:
We can observe that if $x\subset y$ and $y\subset x$
It is possible only when $x=y$
As this relation satisfies all the conditions of anti-symmetry, so it is true for Anti-symmetry.
For Transitivity:
If $x\subset y$ , $y\subset z$
$\Rightarrow x\subset z$
As this relation satisfies all the conditions of transitivity, so it is true for Transitivity.
As, we can see that it holds all the given relation holds all the three properties: Reflexivity, Anti-symmetry and Transitivity.
Hence, the correct option is $4$

Note: A function can be defined as a relation in whom no two ordered pairs have the same first element (i.e. there should be only one output for each given input). Types of functions are: One-one function and Many-one function. The most important thing is that all functions are relations but not all relations are functions.