Question

Let X be a family of sets and R be a relation on X defined as ‘A is disjoint from B’. Then R is
(a) Reflexive
(b) Symmetric
(c) Anti – symmetric
(d) Transitive.

Answer 1

Hint: We solve this problem by checking the given relation whether it satisfies reflexive, symmetric, and antisymmetric relations.
We use the condition that if A is disjoint with B then
\[A\cap B=\phi \]
We use the notation of relation that if A related to B in relation R as \[ARB\]
(1) A relation R is said to be reflexive if \[ARA\] satisfies the given relation.
(2) If \[ARB\] satisfies the relation and \[BRA\] satisfies the relation then that relation is said to be symmetric
(3) A relation which is not symmetric is called antisymmetric
(4) If \[ARB\] and \[BRC\] satisfies the relation then \[ARC\] also satisfies the relation then that relation is said to transitive.

Complete step by step answer:
We are given that the relationship as ‘A is disjoint from B’
We know that the condition that if A is disjoint with B then
\[A\cap B=\phi \]
So, we can say that the given relation is that intersection of two sets is an empty set.
(1) Now, let us check whether the given relation is reflexive or not.
We know that relation R is said to be reflexive if \[ARA\] satisfies the given relation.
Let us check the intersection of A with itself that is
\[\Rightarrow A\cap A=A\ne \phi \]
Here, we can see that the relation on A that is \[ARA\] doesn’t satisfy the given relation because the intersection of A with itself is not an empty set.
Therefore we can conclude that the given relation is not reflexive.
(2) Now, let us check whether it is a symmetric relation.
We know that if \[ARB\] satisfies the relation and also \[BRA\] satisfies the relation then that relation is said to be symmetric
Let us assume that \[ARB\] satisfies the given relation then we get
\[\Rightarrow A\cap B=\phi \]
We know that the intersection is commutative that is
\[A\cap B=B\cap A\]
By using this condition we can say that
\[\Rightarrow B\cap A=\phi \]
Here, we can see that \[BRA\] also satisfies the given relation
Therefore we can conclude that the given relation is symmetric.
(3) Now, let us check whether the given relation satisfies transitive
We know that if \[ARB\] and \[BRC\] satisfy the relation then \[ARC\] also satisfy the relation then that relation is said to transitive.
Let us assume that \[ARB\] and \[BRC\] holds for the given relation then we get
\[\begin{align}
  & \Rightarrow A\cap B=\phi \\
 & \Rightarrow B\cap C=\phi \\
\end{align}\]
Here, we can say that \[A\cap C\ne \phi \]
Let us take one example of A, B and C as follows.
\[A=\left\{ 1,2,3,4 \right\},B=\left\{ 5.6 \right\},C=\left\{ 3,4,8 \right\}\]
Here, we can see that
\[\begin{align}
  & \Rightarrow A\cap B=\phi \\
 & \Rightarrow B\cap C=\phi \\
\end{align}\]
But the intersection of set A and set C is
\[\Rightarrow A\cap C=\left\{ 3,4 \right\}\ne \phi \]
Therefore we can conclude that the given relation is not transitive.
So, we can say that the given relation satisfies only symmetric.
Therefore, option (b) is the correct answer.

Note:
 Students may do mistakes in the transitive relation.
We know that if \[A=B,B=C\] then we can conclude that \[A=C\]
But, we cannot use this formula to sets that is if \[A\cap B=B\cap C=k\] then \[A\cap C\] may or may not equal to \[k\]
It happens the same times and may not happens some times
So, we cannot conclude directly from the relation that if \[A=B, B=C\] then we can conclude that \[A=C\]