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Show that each of the relation R in the set A = { x ∈Z: 0≤...

Show that each of the relation R in the | Class 12 Mathematics Chapter Relations and Functions, Relations and Functions NCERT Solutions

Question: Show that each of the relation R in the set A = { x ∈Z: 0≤x≤12}, A={x} given by
(i) R = { (a,b) : |a - b| is a multiple of 4}
(ii) R = {(a,b):a = b} is an equivalence relation.
Find the set of all elements related to 1 in each case.

Answer:

A = { x ∈ Z : 0 ≤ x ≤ 12} = {0,1,2,3,4,5,6,7,8,9,10,11,12}

(i) R = { (a,b) : |a - b| is a multiple of 4}

For any element a ∈A, we have (a, a) ∈ R as |a - a = 0|is a multiple of 4.

∴R is reflexive.

Now, let (a, b) ∈ R ⇒ |a - b| is a multiple of 4.

⇒ |-(a - b)| = ⇒ |b - a| is a multiple of 4.

⇒ (b, a) ∈ R

∴R is symmetric.

Now, let (a, b), (b, c) ∈ R.

⇒ |(a - b)| is a multiple of 4 and |(b - c)| is a multiple of 4.

⇒ (a - b) is a multiple of 4 and (b - c) is a multiple of 4.

⇒ (a - c) = (a – b) + (b – c) is a multiple of 4.

⇒ |a - c| is a multiple of 4.

⇒ (a, c) ∈R

∴ R is transitive.

Hence, R is an equivalence relation.

The set of elements related to 1 is {1, 5, 9} since

|1 - 1| = 0 is a multiple of 4,

|5 - 1| = 4 is a multiple of 4, and

|9 - 1| = 8 is a multiple of 4.

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

For any element a ∈A, we have (a, a) ∈ R, since a = a.

∴R is reflexive.

Now, let (a, b) ∈ R.

⇒ a = b

⇒ b = a

⇒ (b, a) ∈ R

∴R is symmetric.

Now, let (a, b) ∈ R and (b, c) ∈ R.

⇒ a = b and b = c

⇒ a = c

⇒ (a, c) ∈ R

∴ R is transitive.

Hence, R is an equivalence relation.

The elements in R that are related to 1 will be those elements from set A which are equal to 1.

Hence, the set of elements related to 1 is {1}.

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Comments

Aithihya
2019-07-01 22:30:11

It was helpful

Parul
2019-04-13 10:30:55

Thanks it was helpful and I needed this urgently thanks a lot for the helpðð

Robin
2018-09-04 20:01:20

Attt sirraaaa fudduðð

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Welcome to the NCERT Solutions for Class 12 Mathematics - Chapter . This page offers a step-by-step solution to the specific question from Excercise 1 , Question 9: Show that each of the relation R in the set A = { x ∈Z: 0≤x≤12}, A={x} given by (....

Show that each of the relation R in the | Class 12 Mathematics Chapter Relations and Functions, Relations and Functions NCERT Solutions

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