Relationship: reflexive, symmetric, antisymmetric, transitive

In summary, the relation R on all integers where aRy is |a-b|<=3 is reflexive and symmetric, but not antisymmetric or transitive.
  • #1
nicnicman
136
0

Homework Statement


Determine which binary relations are true, reflexive, symmetric, antisymmetric, and/or transitive.

The relation R on all integers where aRy is |a-b|<=3

Homework Equations


The Attempt at a Solution


The relationship is reflexive because any number minus itself will be zero which is less than 3.
The relationship is symmetric because whenever |a-b|<=3 then |b-a|<=3 is also true.
The relationship is not antisymmetric. Consider (2, 1) and (1, 2). Int his case, |2-1|<=3 and |1-2|<=3.
The relationship is not transitive. Consider (4, 1) and (1, 0). |4-1|<=3 and |1-0|<=3, but |4-0| is not <= 3.

I just want to make sure I'm understanding this correctly. Is this right or am I missing something? I can't think of a case where the relationship wouldn't be symmetric.
 
Last edited:
Physics news on Phys.org
  • #2
nicnicman said:

Homework Statement


Determine which binary relations are true, reflexive, symmetric, antisymmetric, and/or transitive.

The relation R on all integers where aRy is |a-b|<=3


Homework Equations





The Attempt at a Solution


The relationship is reflexive because any number minus itself will be zero which is less than 3.
The relationship is symmetric because whenever |a-b|<=3 then |b-a|<=3 is also true.
The relationship is not antisymmetric. Consider (2, 1) and (1, 2). Int his case, |2-1|<=3 and |1-2|<=3.
The relationship is not transitive. Consider (4, 1) and (1, 0). |4-1|<=3 and |1-0|<=3, but |4-0| is not <= 3.

I just want to make sure I'm understanding this correctly. Is this right or am I missing something? I can't think of a case where the relationship wouldn't be symmetric.

Those all look correct to me.
 
  • #3
Thank you for the reassurance.
 

1. What does it mean for a relationship to be reflexive?

A reflexive relationship is one where every element in the set is related to itself. In other words, every element has a connection to itself in the relationship.

2. Can a relationship be both symmetric and antisymmetric?

No, a relationship cannot be both symmetric and antisymmetric at the same time. A symmetric relationship means that if element A is related to element B, then element B is also related to element A. On the other hand, an antisymmetric relationship means that if element A is related to element B, then element B cannot be related to element A. These two properties contradict each other, so a relationship cannot have both.

3. How does transitivity affect a relationship?

Transitivity means that if element A is related to element B and element B is related to element C, then element A is also related to element C. This property can help determine the overall structure and connections within a relationship.

4. What is an example of a reflexive relationship?

An example of a reflexive relationship is the "is equal to" relationship between numbers. Every number is equal to itself, making this relationship reflexive.

5. How can we determine if a relationship is reflexive, symmetric, antisymmetric, or transitive?

To determine the properties of a relationship, we can examine the elements within the relationship and see if they follow the definitions of reflexive, symmetric, antisymmetric, and transitive. We can also use specific examples and counterexamples to test these properties.

Similar threads

  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
2K
  • Calculus and Beyond Homework Help
Replies
17
Views
10K
  • Calculus and Beyond Homework Help
Replies
5
Views
11K
  • Set Theory, Logic, Probability, Statistics
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
128
  • Calculus and Beyond Homework Help
Replies
5
Views
3K
  • Set Theory, Logic, Probability, Statistics
Replies
7
Views
1K
  • Calculus and Beyond Homework Help
Replies
5
Views
355
Replies
9
Views
646
Back
Top