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

Hint: 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.