Answer to Question #202038 in Discrete Mathematics for zain ul abdeen

Question

Q1:

Determine whether the relation R on the set of all people is reflexive, symmetric,

antisymmetric, and/or transitive, where (a,b) ∈ R if and only if

a) a is taller than b.

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

c) a has the same first name as b.

d) a and b have a common grandparent.

Answer 1

(a)  a is taller than b.

Reflexive : no (is anyone taller than themselves? No)

Symmetric : no(if x is taller than y does that mean that y is taller than x? No)

Antisymmetric: Yes(this relationship never shows symmetry and therefor never shows symmetry on two elements that are different)

Transitive: Yes(if x is taller than y and y is taller than z ,does that mean x is taller than z ? Yes Always)

(b)a and b were born on the same day.

Reflexive: Yes (everyone is born on the same day as themselves)

Transitive: Yes(if x is born on the same day as y then y was born on the same day as x)

Antisymmetric: No(two different people can be born on the same day)

Transitive: Yes(if x is born on the same day as y who was born on the same day as z then was x born on same day as z? Yes)

(c)a has the same first name as b.

Reflexive: Yes(everyone has the same name as themselves)

Symmetric: Yes (if x has same name as y then y must have same first name as x)

Antisymmetric: No(two different people can have same name)

Transitive: Yes (if x has same name as y who has the same first name as z then does x have same first name as z? Yes)

(d) a and b have a common grandparent.

Reflexive: Yes (everyone has the same grandparent as themselves)

Symmetric: Yes (if x has same grandparent as y then y must have same grandparent as x)

Antisymmetric: No (two different people can have same grandparent)

Transitive: No (You and your cousin can share a grandparent and your cousin can share their other grandparent with someone else. You will not share a grandparent with that someone else )