Question

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.

Answer 1

a) R(a,b) means a is taller than b

• Not Reflexive: since R(a,a) doesn't hold as a is not taller than a
• Not Symmetric: since if R(a,b) then R(b,a) does not hold as if a is taller than b then b cannot be taller than a.
• Antisymmetric: since if R(a,b) with a not equals b then R(b,a) must not hold. This is true because for two distinct people a and b, if a is taller than b then b is not taller than a.
• Transitive: since if R(a,b) and R(b,c) then R(a,c) must hold. This is true because if a is taller than b and b is taller than c then a is taller than c.

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

• Reflexive: since R(a,a) holds as for a person a, it is true that a and a were born on the same day.
• Symmetric: since if R(a,b) then R(b,a) holds. This is true because a and b were born on the same day is same as b and a were born on the same day.
• Not Antisymmetric: since R is symmetric as above, it can't be Antisymmetric.
• Transitive: if R(a,b) and R(b,c) then R(a,c) holds. This is true because if a and b are born on the same day and b and c are born on the same day then a and c are also born on the same day.

Feel free to ask any doubt you may have. Happy to help.