Question

Let N be the set of natural numbers greater than 100. Define the relation R by: R = {(x, y) = N*N: the numbers x and y have atleast two common divisors). Then Ris
a.Reflexive, Symmetric and transitive relation
b.Symmetric, transitive and NOT Reflexive relation
c.Reflexive, transitive and NOT Symmetric relation
d. Reflexive Symmetric and NOT transitive relation​

Answer 1

Answer:

If R is a relation on set N, then for it to be reflexive,

(a,a)∈R for all a∈N. But (4,4)∈

/

R. Hence not reflexive.

If greatest common divisor of a and b is 2,

then greatest common divisor of b and a is also 2.

So, the given relation is symmetric. (∵aRb⇒bRa).

The given relation is not transitive as (6,10)∈R and (10,18)∈R, but (6,18)∈

/

R (greatest common divisor of 6 and 18 is 6)

Video Explanation

Answer 2

Answer:

$x^3 – 3x^2 y – xy^2 + 3y^3 = 0 \\ \\ </p><p>⇒ x(x^2 – y^2 ) – 3y (x^2 – y^2 ) = 0$

⇒ (x – 3y) (x – y) (x + y) = 0

⇒ Now, x = y ∀ ∀ (x, y) ∈ N × × N so reflexive But not symmetric & transitive

See, (3,1) satisfies but (1,3) does not. Also (3,1) & (1,-1) satisfies but (3,-1) does not.