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For α ∈ N, consider a relation R on N given by R = {(x, y) : 3x + αy is a multiple of 7}. The relation R is an equivalence relation if and only if
  • a)
    α = 14
  • b)
    α is a multiple of 4
  • c)
    4 is the remainder when α is divided by 10
  • d)
    4 is the remainder when α is divided by 7
Correct answer is option 'D'. Can you explain this answer?
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For α ∈ N, consider a relation R on N given by R = {(x, y) ...
For R to be reflexive ⇒ xRx
⇒ 3x + αx = 7x ⇒ (3 + α)x = 7K
⇒ 3 + α = 7λ ⇒ α = 7λ - 3 = 7N + 4, K, λ, N ∈ I
 When α is divided by 7, the remainder is 4.
For R to be symmetric, xRy⇒  yRx
3x + αy = 7N1, 3y + αx = 7N2
⇒ (3 + α)(x + y) = 7(N1 + N2) = 7N3, which holds when 3 + α is multiple of 7.
 α = 7N + 4
For R to be transitive,
xRy & yRz ⇒ xRz
3x + αy = 7N1 & 3y + αz = 7N2
And 3x + αz = 7N3
∴ 3x + 7N2 - 3y = 7N3
∴ 7N1 - αy + 7N2 - 3y = 7N3
 ∴ 7(N1 + N2) - (3 + α)y = 7N3
 ∴ (3 + α)y = 7N, which is true again; when 3 + α is divisible by 7, i.e. when α divided by 7, the remainder is 4.
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For α ∈ N, consider a relation R on N given by R = {(x, y) ...
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For α ∈ N, consider a relation R on N given by R = {(x, y) : 3x + αy is a multiple of 7}. The relation R is an equivalence relation if and only ifa)α = 14b)α is a multiple of 4c)4 is the remainder when α is divided by 10d)4 is the remainder when α is divided by 7Correct answer is option 'D'. Can you explain this answer?
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For α ∈ N, consider a relation R on N given by R = {(x, y) : 3x + αy is a multiple of 7}. The relation R is an equivalence relation if and only ifa)α = 14b)α is a multiple of 4c)4 is the remainder when α is divided by 10d)4 is the remainder when α is divided by 7Correct answer is option 'D'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about For α ∈ N, consider a relation R on N given by R = {(x, y) : 3x + αy is a multiple of 7}. The relation R is an equivalence relation if and only ifa)α = 14b)α is a multiple of 4c)4 is the remainder when α is divided by 10d)4 is the remainder when α is divided by 7Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For α ∈ N, consider a relation R on N given by R = {(x, y) : 3x + αy is a multiple of 7}. The relation R is an equivalence relation if and only ifa)α = 14b)α is a multiple of 4c)4 is the remainder when α is divided by 10d)4 is the remainder when α is divided by 7Correct answer is option 'D'. Can you explain this answer?.
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