relation

Question

A binary relation R on Z × Z is defined as follows:
                                                        (a, b) R (c, d) iff a = c or b = d
Consider the following propositions:
1. R is reflexive.                                       2. R is symmetric.
3. R is antisymmetric.
Which one of the following statements is True?

A Both 1 and 2 are true 

B 1 is true and 2 is false

 C 1 is false and 3 is true

D Both 2 and 3 are true

Comments

Yes that’s what I am saying dont check internally between ab and cd.

Normally we check for symmetric relation, xRy then yRx so here x = (a,b) and y = (c,d)

now take a=c such as (1,2) R (1,5) now here x= 1,2 and y =1,5

now reverse the whole thing yRx = (1,5) R (1,2) still a=c preserved and symmetric .

hope it will be clear now :)

---

thnku : )

---

@ashwin do you think it's transitive also?

Answer 1

(a) : R is reflexive: Since (a, b) R (a, b) for all elements (a, b) because a = a and b = b are always true

(b): R is symmetric: Since (a, b) R (c, d) and a = c or b = d which can be written as c = a or d = b. So, (a, b) R (a, b) is true 

(c): R is not antisymmetric: Since (1, 2) R (1, 3) and 1 = 1 or 2 = 3 true b/c 1 = 1. So (1, 3) R (1, 2) but here 2 ≠ 3 so (1, 2) ≠ (1, 3)

. So, only statement 1 and 2 are correct.