Set, Relation and FunctionHard
Question
Let $A = \{ 0,1,2,\ldots.9)$. Let R be a relation on A defined by $(x,y) \in R$ if and only if $|x - y|$ is a multiple of 3.
Given below are two statements:
Statement I: $n(R) = 36$
Statement II: $R$ is an equivalence relation.
In the light of the above statements, choose the correct answer from the options given below
Options
A.Both Statement I and Statement II are correct
B.Statement I is incorrect but Statement II is correct
C.Statement I is correct but Statement II is incorrect
D.Both Statement I and Statement II are incorrect
Solution
Number of form $3\text{ }K = 4$
Number of form $3\text{ }K + 1 = 3$
Number of form $3\text{ }K + 2 = 4$
$4 \times 4 + 3 \times 3 + 3 \times 3 = 34$ relations
$${\Rightarrow xRy \Rightarrow yRx }{\Rightarrow (x - y) = 3\lambda,(y - z) = 3\mu }{\Rightarrow (x - z) = 3(\lambda + \mu) }$$R is reflexive, symmetric and transitive $S_{2}$ is ture Ans. $S_{1}$ is false but $S_{2}$ is ture
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