relation on A that is symmetric and transitive but not reflexive

In summary, the relation {(1,2),(2,1),(1,1)} on set A = {1,2,3,4} is symmetric and transitive, but not reflexive. The mistake in the attempt at a solution was that 2R1 and 1R2 imply 2R2, but (2,2) is not an element of set A.
  • #1
Easy_as_Pi
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Homework Statement


Let A = {1,2,3,4}. Give an example of a relation on A that is symmetric and transitive, but not reflexive.

Homework Equations


Symmetric: if aRb then bRa
Transitive: if aRb and bRc then aRc
Reflexive: aRa for all a in A

The Attempt at a Solution


{(1,2),(2,1),(1,1)} It's symmetric because 1R2 and 2R1. Not reflexive because (2,2)...(4,4) are not elements and transitive because 1R2 and 2R1 so 1R1. Yet, this one got marked wrong on my homework. I'm going to assume my teacher is right, and I'm wrong. Can anyone find my mistake?
 
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  • #2


2R1 and 1R2 implies 2R2 if your relation is transitive. But (2,2) isn't in A. So your teacher is right.
 
  • #3


Oh wow, I can't believe I missed that! Thanks for that!
 

1. What does it mean for a relation on set A to be symmetric?

A relation on set A is symmetric if for every pair of elements (a,b) in A, if (a,b) is in the relation, then (b,a) is also in the relation.

2. Can a relation on set A be both symmetric and transitive, but not reflexive?

Yes, it is possible for a relation on set A to be symmetric and transitive, but not reflexive. This means that the relation follows the properties of symmetry and transitivity, but does not include all elements in A that are related to themselves.

3. How can you tell if a relation on set A is symmetric and transitive, but not reflexive?

To determine if a relation on set A is symmetric and transitive but not reflexive, you can check each individual element in A to see if it is related to itself. If there is at least one element that is not related to itself, then the relation is not reflexive.

4. What is an example of a relation on set A that is symmetric and transitive, but not reflexive?

An example of a relation on set A that is symmetric and transitive, but not reflexive is the "is equal to" relation. In this relation, if two elements are equal, then they are also related to each other. However, not all elements in A are equal to themselves.

5. Why is it important for a relation on set A to be reflexive?

A relation on set A being reflexive is important because it ensures that every element in A is related to itself. This helps to establish a sense of equality and helps to define the relation more clearly. Additionally, many important mathematical concepts and theorems rely on reflexive relations.

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