Let A={1,2,3,...9} and R be the relation in A×A defined by (a,b) R (c,d) if a+b=b+c for (a,b),(c,d) in A×A. Prove that R is an equivalence relation and also obtain the equivalent class [(2,5)].

Given that A={1,2,3,...9}(a,b) R (c,d) if a+b=b+c for (a,b),(c,d) in A×A.(c,d)∈A×A Let (a,b)R(a,b)⇒ a+b=b+a,∀a,b∈Awhich is true for any a,b∈AHence, R is reflexiveLet (a,b)R∈(c,d)a+d=b+cc+b=+a⇒(c,d)R(a,b)So, R is symmetricLet (a,b)R(c,d) and (c,d)R(e,f)a+d=b+c and c+f=d+ea+d=b+c and d+e=c+f(a+d)−(d+e)=(b+c)−(c+f)(a−e)=b−fa+f=b+e(a,b)R(e,f)Now, equivalence class containing [(2,5) is {(1,4),(2,5),(3,6),(4,7),(5,8),(6,9)}]{(2,5)} is {(1,4),(2,5),(3,6),(4,7),(5,8),(6,9)}

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Let A={1,2,3,...9} and R be the relation in A A defined by (a,

Question

Let $A = {1, 2, 3, ...9}$ and $R$ be the relation in $A \times A$ defined by $(a, b) R (c, d)$ if $a + b = b + c$ for $(a, b), (c, d)$ in $A \times A$ . Prove that $R$ is an equivalence relation and also obtain the equivalent class $[(2, 5)]$ .

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Let

A = {1, 2, 3, ...9}

and

R

be the relation in

A \times A

defined by

(a, b) R (c, d)

a + b = b + c

for

(a, b), (c, d)

A \times A

. Prove that

R

is an equivalence relation and also obtain the equivalent class

[(2, 5)]

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Text solutionVerified

Given that

A = {1, 2, 3, ...9} (a, b) R (c, d)

a + b = b + c

for

(a, b), (c, d)

A \times A

(c, d) \in A \times A

Let

(a, b) R (a, b)

\Rightarrow a + b = b + a, \forall a, b \in A

which is true for any

a, b \in A

Hence,

R

is reflexive

Let

(a, b) R \in (c, d)

a + d = b + c

c + b = + a \Rightarrow (c, d) R (a, b)

So,

R

is symmetric

Let

(a, b) R (c, d)

and

(c, d) R (e, f)

a + d = b + c

and

c + f = d + e

a + d = b + c

and

d + e = c + f (a + d) - (d + e) = (b + c) - (c + f)

(a - e) = b - f

a + f = b + e

(a, b) R (e, f)

Now, equivalence class containing

[(2, 5) i s {(1, 4), (2, 5), (3, 6), (4, 7), (5, 8), (6, 9)}]

{(2, 5)}

{(1, 4), (2, 5), (3, 6), (4, 7), (5, 8), (6, 9)}

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Question Text	Let $A = {1, 2, 3, ...9}$ and $R$ be the relation in $A \times A$ defined by $(a, b) R (c, d)$ if $a + b = b + c$ for $(a, b), (c, d)$ in $A \times A$ . Prove that $R$ is an equivalence relation and also obtain the equivalent class $[(2, 5)]$ .
Answer Type	Text solution:1
Upvotes	150

Let A={1,2,3,...9} and R be the relation in A×A defined by (a,b) R (c,d) if a+b=b+c for (a,b),(c,d) in A×A. Prove that R is an equivalence relation and also obtain the equivalent class [(2,5)].

Text solutionVerified

Practice questions from similar books

Let $A = {1, 2, 3, ...9}$ and $R$ be the relation in $A \times A$ defined by $(a, b) R (c, d)$ if $a + b = b + c$ for $(a, b), (c, d)$ in $A \times A$ . Prove that $R$ is an equivalence relation and also obtain the equivalent class $[(2, 5)]$ .