Let S = \ 1,2,3, ... ,10\. Suppose M is the set of all the subsets of S, then the relation R = \(A,B) : A intersection B ≠ phi;A,B in M\ is:

(d): Given, A in M => A intersection A = phi, if A = phi \ So, R is not reflexive. If (A,B) in R => A intersection B ≠ phi => B intersection A ≠ phi => (B,A) in R So, R is symmetric. If (A,B) and (B,C) in R => A intersection B ≠ phi and B intersection C ≠ phi => A intersection C is not necessarily non-empty set. Hence, A intersection C not necessarily belongs to R. So, R is not a transitive relation. Therefore Given relation is symmetric and reflexive only.

Let S = { 1,2,3,...,10}. Suppose M is the set of all th... | Relation (Mathematics)

(d): Given, $A \in M \Rightarrow A \cap A = \phi$ , if $A = \phi \\$ So, $R$ is not reflexive. $\\$ If $(A,B) \in R \Rightarrow A \cap B \neq \phi \Rightarrow B \cap A \neq \phi \Rightarrow (B,A) \in R \\$ So, $R$ is symmetric. $\\$ If $(A,B)$ and $(B,C) \in R \Rightarrow A \cap B \neq \phi$ and $B \cap C \neq \phi$ $\\$ $\Rightarrow A \cap C$ is not necessarily non-empty set. Hence, $A \cap C$ not necessarily belongs to $R$ . $\\$ So, $R$ is not a transitive relation. $\\$ $\therefore$ Given relation is symmetric and reflexive only.

Explanation

The minimum number of elements that must be added to the relation R = {(a,b),(b,c),(b,d)} on the set { a,b,c,d} so that it is an equivalence relation is

Let R be the set of real numbers. end{enumerate} Statement-1: A = {(x,y) in R X R:y - x is an integer } is an equivalence . relation on R. Statement-2: B = {(x,y) in R X R:x = alpha y for some rational number } is . an equivalence relation on R.

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Let A = { 0,3,4,6,7,8,9,10} and R be the relation defined on A such that R = {(x,y) in A X A:x - y is odd positive integer or x - y = 2}. The minimum number of elements that must be added to the relation R, so that it is a symmetric relation, is equal to .

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