Question

Determine whether the relation R on the set of all people is irreflexive, where (a, b) ∈ R if and
only if :
i) a and b were born on the same day.
ii) a has the same first name as b.
iii) a and b have a common grandparent

Answer 1

A relation $R$ on the set $A$ is irreflexive if for every $a\in A, (a, a)\not\in R.$ That is, $R$ is irreflexive if no element in $A$ is related to itself.

i) $a$ and $b$ were born on the same day.

Reflexive since one is born on one’s own birthday.

A relation is not irreflexive.

ii) $a$ has the same first name as $b.$

Assuming everyone has a first name, then it’s reflexive.

A relation is not irreflexive.

iii) $a$ and $b$ have common grandparents.

Assume this means $\exist g$ such that $g$ is a grandparent of both $a$ and $b.$ Then $R$ is reflexive (everyone has the same grandparent as themselves).

A relation is not irreflexive.