Examples of Identifying Properties in Relations
Is the following relation reflexive, irreflexive, symmetric,
antisymmetric, or transitive? R = {(a,b) | a,b
∈
Z
+
∧
a, 2a, and b
are side lengths of a triangle} Note: For all triangles, the sum
of the lengths of any two sides must exceed the length of the
third side.
Reflexive? No – because (a,a)
∉
R, this is because a triangle can not
have sid lengths, a, a and 2a.
Irreflexive? Yes – the previous argument holds for all positive
integers a.
Symmetric? No – (a, 2a)
∈
R, since we can have a triangle with side
lengths a, 2a and 2a. However, (2a, a)
∉
R because we can not have
a triangle with side lengths 2a, 4a and a.
Antisymmetric? Yes – If we have a
≥
b, then we have (a,b)
∉
R. To
prove this, consider a forming a triangle with side lengths a, 2a,
and b. We know that we must have a+b > 2a for a triangle to be
formed. BUT, a+b
≤
a+a = 2a, which means that a+b is NOT
greater than 2a. Thus, in this situation, we have (a,b)
∉
R. Thus, for
any element (a,b)
∈
R, we must have a < b. For each of these
elements, we can guarantee that (b,a)
∉
R since b > a.
Transitive? No – (a, 2a)
∈
R as shown above, and we also know that
(2a, 4a)
∈
R, by a similar analysis. But, we can show that (a,4a)
∉
R
because a triangle can not have side lengths a, 2a and 4a.