Thread: Relations help

For each of the following relations on the set of all people, determine whether it is reflexive, symmetric, antisymmetric,

and/or transitive. (Justification expected.)

a) The relation Q where (a, b)EQ if and only if a and b were born on the same day.
b) The relation R where (a, b)ER if and only if a is taller than b.

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Originally Posted by thehollow89

For each of the following relations on the set of all people, determine whether it is reflexive, symmetric, antisymmetric,

and/or transitive. (Justification expected.)

a) The relation Q where (a, b)EQ if and only if a and b were born on the same day.

Test for reflexive: Let b = a.

Is person a born on the same day as itself? I certainly think so. Therefore, Q is reflexive.


Test for symmetric: If (a,b), then (b,a).

If person a is born on the same day as person b, is it true that person b is born on the same day as person a? This is true. Therefore, Q is symmetric.


Test for transitive: If (a,b) and (b,c), then (a,c).

If person a is born on the same day as person b, and person b is born on the same day as person c, is it true that person a is born on the same day as person c? Again, this is true. Therefore, Q is transitive.

b) The relation R where (a, b)ER if and only if a is taller than b.

Reflexive: is (a,a) true?

Is person a taller than itself? Well, people have one height at any given time, so I don't think this will work. R is not reflexive.


Symmetric: if (a,b) then (b,a)

If person a is taller than person b, is person b taller than person a? Well, a is taller, then a can not also be shorter. R is not symmetric.


Transitive: if (a,b) and (b,c), then (a,c)

If person a is taller than person b, and person b is taller than person c, is person a taller than person c? Seems that the heights of a,b,c can be ranked as a>b>c, so a>c. R is transitive.