Question

The relation ≤ 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 $a=b\forall a,b\in R$ (Antisymmetry)
(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 $p\left( A \right)$ has?
(a). (i) and (ii)
(b). (i) and (iii)
(c). (ii) and (iii)
(d). (i), (ii) and (iii)

Answer 1

Hint: First observe the relation given in question. Find all the properties related to the relation. Try to substitute values or variables to check their properties. Try to use definitions of Reflexive, transitive, symmetric, anti symmetry. First and the foremost thing to do is use the definition of relation and say that the given relation satisfies all the conditions needed for it to be a relation. Next see the range on which relation is defined. While checking properties, substitute a few elements only which satisfy the range given in the question.

Complete step-by-step anwer:

Relation:- The relations in maths are also called as binary relation over se x,y is a subset of the Cartesian product of the set x,y that is the relation contains the elements of set XxY. XxY, is a set of ordered pairs consisting of element x in X and y in Y. It encodes the information, $\left( x,y \right)$ will be in relation set if and only if x is related y in the relation property given.
Reflexive relation:- A relation R is said to be reflexive, if a is an element in the range. If the pair $\left( a,a \right)$ belongs to relation then relation is reflexive. In other words, if a is related to itself by a given relation then relation is reflexive.
Symmetric Relation:- A relation R is said to symmetric, if a,b are elements in the range. If the pair $\left( a,b \right)$ belongs to relation then $\left( b,a \right)$ must belong to relation. In other words if a is related to b, then b must also be related to a for a relation to be symmetric.
Transitive Relation:- If $\left( a,b \right)$ belongs to relation and $\left( b,c \right)$ also belongs to the relation, then $\left( a,c \right)$ must belong to relation to make it transitive.
Anti-symmetric Relation:- If $\left( a,b \right)$ and $\left( b,a \right)$ are given belong to relation, then a must be equal to b to make the relation as anti symmetric. If a relation is reflexive, Transitive and Symmetric, then it is an equivalence relation
Given relation $R=\le$
$a\le a$satisfies, So, $\left( a,a \right)\in R$ it is Reflexive.
$a\le b$ and $b\le a\Rightarrow a=b$ . It is Antisymmetric.
$a\le b$ , $b\le c\Rightarrow a\le c$ .It is Transitivity.
They asked us about the Relation $\subset$ on $p\left( A \right)$.
Power set: In mathematics, the power set of any set S is a set of all the subsets of S including the empty set $\left( \varnothing \right)$ . It is denoted as $p\left( S \right)$ .
So, Here $p\left( A \right)$ implies all sets possible because a ≥ all the numbers.
A set A is a subset of itself. $A\subset A$ .So, it is Reflexive.
If $A\subset B$ and $B\subset A$ ,$\Rightarrow A$ has all elements of B and B has all elements $A\Rightarrow A=B$ Anti symmetric.
$A\subset B$ , $B\subset C\Rightarrow A\subset C$. It is transitive.
So, (i), (ii), (iii), the relation $\subset$ shows all the given properties.
So, Option (D) is correct answer.

Note: Use the given statements in case you forgot the definitions of reflexive, anti -symmetric, transitive because it is given in the statements that it is reflexive or not, anti –symmetric or not, transitive or not from that one can derive the definition on one’s own. Observe that the relation $\subset$ is not symmetric (point to note). By this we can say that the relation $\subset$ is not an equivalence relation.