Relevant definitions:
Definition: “A relation R on a set A is called symmetric if (b, a) ∈ R whenever (a,b) ∈ R, for all a,b ∈ A. A relation R on a set A such that for all a, b ∈ A, if (a, b) ∈ R and (b, a) ∈ R, then a = b is called antisymmetric.” Source: Discrete Mathematics and its Applications by Rosen.
Now, let’s solve the exercise.
a) both symmetric and antisymmetric
R={(a,b)| a=b}
Symmetric: if (a,b) ∈ R, it follows that a=b, then (b, a) ∈ R. Therefore, the relation R is symmetric.
Antisymmetric: for all a, b ∈ A, if (a, b) ∈ R, and (b, a) ∈ R, then a = b. Therefore, it is antisymmetric.
b) neither symmetric nor antisymmetric
R={(a,b)| a is a multiple of b}
Symmetric: (2,4) ∈ R, (4,2) ∉ R. Therefore, the relation R is not symmetric.
Antisymmetric: (2,-2) ∈ R, (-2,2) ∈ R, and 2≠-2. Therefore, it is not antisymmetric.
Related exercises:
- Show that the relation R = ∅ on a nonempty set S is symmetric and transitive, but not reflexive
- Determine whether the relation R on the set of all real numbers is reflexive, symmetric, antisymmetric, and/or transitive, where (x,y)∈R if and only if
- a) List all the ordered pairs in the relation R = {(a,b) | a divides b} on the set {1,2,3,4,5,6}. b) Display this relation graphically, as was done in Example 4. c)Display this relation in tabular form, as was done in Example 4
- Show that the relation R=∅ on the empty set S=∅ is reflexive, symmetric, and transitive
- Determine whether the relation R on the set of all Web pages is reflexive, symmetric, antisymmetric, and/or transitive, where (a, b) ∈ R if and only if
