Consider the following statements in respect of the relation R in the set IN of natural numbers defined by xRy if x2 - 5xy + 4y2 = 0 : 1. R is reflexive 2. R is symmetric 3. R is transitive Which of the above statements is /are correct ?
Consider the following statements in respect of the relation R in the set IN of natural numbers defined by xRy if x2 - 5xy + 4y2 = 0 : 1. R is reflexive 2. R is symmetric 3. R is transitive Which of the above statements is /are correct ? Correct Answer 1 only
Concept:
1. Reflexive: Each element is related to itself.
- R is reflexive if for all x ∈ A, xRx.
2. Symmetric: If any one element is related to any other element, then the second element is related to the first.
- R is Symmetric if for all x, y ∈ A, if xRy, then yRx.
3. Transitive: If any one element is related to a second and that second element is related to a third, then the first element is related to the third.
- R is transitive if for all x, y, z ∈ A, if xRy and yRz, then xRz.
4. R is an equivalence relation if A is nonempty and R is reflexive, symmetric, and transitive.
Calculation:
We have R = {(x,y): x2 - 5xy + 4y2 = 0, x,y ∈ N}.
Statement I: Reflexive
If y = x, then x2 - 5x2 + 4x2 = 0
⇒ (x,x) ∈ R.
⇒ R is reflexive
Statement II: Symmetric
Take (4, 1) i.e. x = 4 and y = 1
We have (4)2 − 5(4)(1) + 4(1)2 = 16 − 20 + 4 = 0
⇒ (4,1) ∈ R.
Also (1)2− 5(1)(4) + 4(4)2 = 1 − 20 + 64 = 45 ≠ 0
⇒ (1,4) ∉ R.
⇒ R is not symmetric.
Statement III: Transitive
(16,4) ∈ R because
(16)2 − 5(16)(4) + 4(4)2 = 256 − 320 + 64 = 0
Also (4,1) ∈ R because
⇒ (4)2 − 5(4)(1) + 4(1)2 = 16 − 20 + 4 = 0
Now, (16,1) ∈ R if (16)2 − 5(16)(1) + 4(1)2 = 0
⇒ 256 − 80 + 4 = 48 ≠ 0, which is not so.
⇒ (16,4), (4,1) ∈ R and (16,1) ∉ R
⇒ R is not transitive.
∴ R is only Reflexive.
