Relationship: reflexive, symmetric, antisymmetric, transitive

In summary, the relation R on P = {a, b, c} where R = {(a, a), (a, b), (a, c), (b, c), (c, b)} is not reflexive, symmetric, antisymmetric, or transitive. It is possible for there to not be any binary relations.
  • #1
nicnicman
136
0

Homework Statement


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

The relation R on P = {a, b, c} where R = {(a, a), (a, b), (a, c), (b, c), (c, b)}


Homework Equations





The Attempt at a Solution


Not reflexive because there is no (b, b) or (c, c).
Not symmetric because there is (a, b), but not (b, a).
Not antisymmetric because there is (b, c) and (c, b).
Not transitive because there is (b, c) and (c, b) but no (b, b).

Is it possible for there to not be any binary relations?
 
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 P = {a, b, c} where R = {(a, a), (a, b), (a, c), (b, c), (c, b)}


Homework Equations





The Attempt at a Solution


Not reflexive because there is no (b, b) or (c, c).
Not symmetric because there is (a, b), but not (b, a).
Not antisymmetric because there is (b, c) and (c, b).
Not transitive because there is (b, c) and (c, b) but no (b, b).

Is it possible for there to not be any binary relations?

It is a binary relation, but as you say, it doesn't have any of those properties.
 
  • #3
Yeah, that's what I thought. Thanks for the help again.
 

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

A reflexive relationship is one in which every element is related to itself. In other words, for all elements x in the set, (x,x) is a member of the relationship. An example of a reflexive relationship is "is equal to", where every element is equal to itself.

2. Can a relationship be both symmetric and antisymmetric?

No, a relationship cannot be both symmetric and antisymmetric. A symmetric relationship is one in which if (x,y) is a member, then (y,x) is also a member. On the other hand, an antisymmetric relationship is one in which if (x,y) is a member, then (y,x) is not a member unless x equals y. These two properties are contradictory, so a relationship cannot possess both at the same time.

3. What is an example of a transitive relationship?

An example of a transitive relationship is "is a subset of". If set A is a subset of set B, and set B is a subset of set C, then it can be inferred that set A is also a subset of set C. This property is useful for making logical conclusions based on a set of given relationships.

4. Can a relationship be both reflexive and irreflexive?

No, a relationship cannot be both reflexive and irreflexive. A reflexive relationship, as mentioned before, has the property that every element is related to itself. On the other hand, an irreflexive relationship has the property that no element is related to itself. These two properties are contradictory, so a relationship cannot possess both at the same time.

5. How can the properties of a relationship be used in real life?

The properties of a relationship, such as reflexivity, symmetry, antisymmetry, and transitivity, can be used to analyze and understand various real life situations. For example, in social networks, the transitive property can be used to predict potential connections between individuals based on their mutual friends. In mathematics and logic, these properties can help to deduce logical conclusions and prove theorems. In general, understanding the properties of relationships can help to better understand and navigate the complex interactions and connections in our daily lives.

Similar threads

  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
17
Views
10K
  • Calculus and Beyond Homework Help
Replies
2
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
10
Views
2K
  • Calculus and Beyond Homework Help
Replies
24
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
7
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
401
  • Calculus and Beyond Homework Help
Replies
5
Views
11K
  • Calculus and Beyond Homework Help
Replies
3
Views
844
  • Calculus and Beyond Homework Help
Replies
7
Views
6K
Back
Top