Let S = {1, 2, 3,…,10}. Suppose M is the set of all the subsets of S, then the relation is:

Question

Let S = {1, 2, 3,…,10}. Suppose M is the set of all the subsets of S, then the relation $R = {(A, B) : A \cap B \neq ϕ; A, B \in M}$ is: [2024]

Smart Achievers · Accepted Answer

(4)

Given, $A \in M \Rightarrow A \cap A = ϕ, if A = ϕ$

So, R is not reflexive.

If $(A, B) \in R \Rightarrow A \cap B \neq ϕ \Rightarrow B \cap A \neq ϕ \Rightarrow (B, A) \in R$

So, R is symmetric.

If $(A, B)$ and $(B, C) \in R \Rightarrow A \cap B \neq ϕ$ and $B \cap C \neq ϕ$

$\Rightarrow A \cap C$ is not necessarily non-empty set.

Hence, $A \cap C$ not necessarily belongs to R.

So, R is not a transitive relation.

$∴$ Given relation is symmetric and reflexive only.

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