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Show that the relation R in the set A={ x Z :0 x 12 }, giv

Question

Show that the relation $R$ in the set
$A = {x \in Z : 0 \leq x \leq 12}$ , given by
$R = {(a, b) : a = b}$ is an equivalence relation. Find the set of all elements related to $1$ in each case.

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Chloe Discussed

Show that the relation

R

in the set

A = {x \in Z : 0 \leq x \leq 12}

, given by

R = {(a, b) : a = b}

is an equivalence relation. Find the set of all elements related to

1

in each case.

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Text solutionVerified

a R a \Rightarrow a = a \forall a \in A

hence

R

is reflexive

a R b \Rightarrow a = b

\Rightarrow b = a \Rightarrow b R a

, hence

R

is symmetric

a R b

and

b R c

\Rightarrow a = b

and

b = c

\Rightarrow a = c, = a R c

hence

R

is transitive
Therefore

R

is an equivalence Relation,

a R 1 \Rightarrow a = 1

Therefore the elements related to

1

{1}

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Question Text	Show that the relation $R$ in the set $A = {x \in Z : 0 \leq x \leq 12}$ , given by $R = {(a, b) : a = b}$ is an equivalence relation. Find the set of all elements related to $1$ in each case.
Answer Type	Text solution:1
Upvotes	150