Let’s start with relevant definitions.
Definition: “A relation R on a set A is called reflexive if (a, a) ∈ R for every element a ∈ A.”
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.”
Definition: “A relation R on a set A is called transitive if whenever (a,b)∈R and (b,c)∈R, then (a,c) ∈ R, for all a,b,c ∈ A.”
The definitions above are from the textbook “Discrete Mathematics and its Applications” by Rosen.
Table of Contents
a)x+y=0
Reflexive: x+x≠0 for all real numbers. Therefore, not reflexive.
Symmetric: if x+y=0, then y+x=0. Therefore, it is symmetric.
Antisymmetric: for x+y=y+x=0, x and y not necessarily are equal. Therefore, it is not antisymmetric.
Transitive: 2 + (-2)=0, (-2)+2=0, but 2+2≠0. Therefore, it is not transitive.
b)x=±y
Reflexive: x=x. Therefore, it is reflexive.
Symmetric: if x=±y, then y==±x. Therefore, it is symmetric.
Antisymmetric(2,-2)∈R, (-2,2)∈R, and 2≠-2. Therefore, it is not antisymmetric.
Transitive: if x=±y and y==±z, then x=±z. Therefore, it is transitive.
c) x−y is a rational number
A number that is expressed as the ratio of two integers, where the denominator should not be equal to zero is rational.
Reflexive: x-x=0 and 0 is rational. Therefore, not reflexive.
Symmetric: if x-y is rational, y-x is also rational. Therefore, it is symmetric.
Antisymmetric: 2-4 is rational, 4-2 is rational, 4≠2. Therefore, it is not antisymmetric.
Transitive: x-y=z is rational, y-t=s is rational, it follows that y=x-z, x-z-t=s, x-t=s+z. The sum of two rational numbers is also rational. Therefore, it is transitive.
d) x=2y
Reflexive: x≠2x if x≠0. Therefore, not reflexive.
Symmetric: if x=2y, then y=x/2, it follows that (y,x)∉R. Therefore, it is not symmetric.
Antisymmetric: if x=2y, then y=2x, it follows that x=y. Therefore, it is antisymmetric.
Transitive: x=2y and y=2z, it follows that x/2=2z, x=4z, (x,z)∉R. Therefore, it is not transitive.
e) xy≥0
Reflexive: x2≥0. Therefore, it is reflexive.
Symmetric: if xy≥0, then yx≥0. Therefore, it is symmetric.
Antisymmetric: (2,-2)∈R, (-2,2)∈R, 2≠-2. Therefore, it is not antisymmetric.
Transitive: (-1,0) ∈R, (0,1)∈R, (-1,1)∉R. Therefore, not transitive.
f) xy=0
Reflexive: x2 equals 0 only when x=0. Therefore, not reflexive.
Symmetric: if xy=0, then yx=0. Therefore, it is symmetric.
Antisymmetric: (2,0)∈R, (0,2)∈R, 2≠0. Therefore, it is not antisymmetric.
Transitive: (-1,0) ∈R, (0,1)∈R, (-1,1)∉R. Therefore, not transitive.
g) x=1
Reflexive: (2,2)∉R. Therefore, not reflexive.
Symmetric: (1,2) ∈R but (2,1)∉R. Therefore, it is not symmetric.
Antisymmetric: if (x,y)∈R and (y,x)∈R, then x=y=1. Therefore, it is antisymmetric.
Transitive: x1,y) ∈R, (y,z)∈R, implies that x=1, it follows that (x,z)∈R. Therefore, it is transitive.
h) x=1 or y=1
Reflexive: (2,2)∉R. Therefore, not reflexive.
Symmetric: (x,y)∈R, implies that x=1 or y=1, it follows that (y,x)∈R. Therefore, it is symmetric.
Antisymmetric: (3,1)∈R, (1,2)∈R, 3≠2. Therefore, it is not antisymmetric.
Transitive: (3,1)∈R, (1,2)∈R, (3,2)∉R. Therefore, not transitive.
Related exercises:
- Give an example of a relation on a set that is a) both symmetric and antisymmetric. b) neither symmetric nor antisymmetric
- Show that the relation R = ∅ on a nonempty set S is symmetric and transitive, but not reflexive
- 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
