Here are some of the discussion examples for the week of October 29 or the week of
November 5
1. Find the properties of the following relation: R = {(a, b)N  N | a + b is even}
Reflexive: Since a + a = 2a for all natural numbers a, R is reflexive.
Symmetric: If (a, b)  R, then a + b = 2k for some integer k.
Then, b + a = 2k also, so (b, a)  R. Thus, R is symmetric.
Clearly, R is not irreflexive, antisymmetric or asymmetric, so the only thing we have left
to consider is the transitive property. Suppose (a, b)  R and (b, c)  R. Then, there
are integers j and k such that
a + b = 2k and b + c = 2j Thus,
a + c + 2b = 2k + 2j
a + c = 2k + 2j -2b
a + c = 2(k + j - b)
Since a + c is even, (a, c)  R and R is transitive.
2. Prove the complement of a symmetric relation is symmetric.
Proof: Let R be a symmetric relation on a set A.
Let (a, b) be an element of R′.
(a, b)  R
Definition of complement
(b, a)  R
Definition of symmetric and R is symmetric
(b, a)  R′
Definition of complement
Therefore R′ is symmetric
Definition of symmetric
3. Prove that if R and S are symmetric relations on set A, then R  S is also symmetric.
Proof: Let R and S be symmetric relations, and let (a, b)  R  S.
Then, (a, b)  R or (a, b)  S
Definition of union
Then, (b, a)  R or (b, a)  S
R and S are symmetric; defn of symmetric
Therefore, (b, a)  R  S
Definition of union
Therefore, R  S is symmetric
Definition of symmetric
4. Prove the inverse of a transitive relation is transitive.
Proof: Let R be a transitive relation on S.
Suppose (a, b)  R-1 and (b, c)  R-1
Then, (b, a)  R and (c, b)  R
Definition of inverse
Thus, (c, a)  R
Given that R is transitive
-1
Therefore, (a, c)  R
Definition of inverse
Therefore R-1 is transitive
Definition of transitive
5. Prove that a relation R on a set S is symmetric if and only if R = R-1
Part 1: If R is symmetric, then R = R-1.
In this case we must show that if R is symmetric, then R  R-1 and R-1  R.
Let (a, b)  R.
Then, (b, a)  R.
Then, (a, b)  R-1
Therefore, R  R-1.
Given that R is symmetric
Definition of inverse
Definition of subset
Let (x, y)  R-1
Then, (y, x)  R
Then, (x, y)  R.
Therefore R-1  R
Therefore R = R-1
Definition of inverse
Given that R is symmetric
Definition of subset
Definition of set equality
Part 2: If R = R-1 then R is symmetric
(Specifically, what we need to show is that if (a, b)  R, then (b, a)  R.)
Let (a, b)  R.
Then, (a, b)  R-1
Then, (b, a)  R.
Therefore, R is symmetric.
Given that R = R-1
Definition of inverse
Definition of symmetric