Is Relation S Reflexive, Symmetric, and Transitive?

• kingstar
In summary, the conversation discusses the relation S defined on the set of people, where xSy holds if person x is taller than person y. The question asks to determine if S is reflexive, symmetric, and transitive, and if it is an equivalence relation. It is concluded that S is not symmetric or reflexive, but it is transitive.
kingstar

Homework Statement

a) Consider the relation S de fiend on the set {t : t is a person} such that xSy holds exactly if
person x is taller than y. Determine if the relation S is reflexive, symmetric and transitive.
Is the relation S an equivalence relation?

Homework Equations

Recall that a relation R de ned on a set A is reflexive if for all x 2 A xRx.
Recall that a relation R de ned on a set A is symmetric if for all x 2 A and y 2 A the xRy implies
yRx.
Recall that a relation R e ned on a set A is transitive if for all x; y; z in A, both xRy and yRz
holds, then xRz holds as well.
Finally recall that a relation R is an equivalence relation if its reflexive, symmetric and transitive.

The Attempt at a Solution

As far as i can see the set is not symmetric or reflexive but I'm not 100% on transitive...
It would be transitive if x > y and y > z then x > z holds...but we aren't given any information on y > z?

So would this set be transitive?

Yes the relationship is not symmetric or reflexive, but it IS transitive.

If ##xRy## holds, then ##x > y##, if ##yRz## holds, then ##y > z##. So person ##x## is taller than person ##y## and person ##y## is taller than person ##z##.

It must be the case that person ##x## must be taller than person ##z##.

Zondrina said:
Yes the relationship is not symmetric or reflexive, but it IS transitive.

If ##xRy## holds, then ##x > y##, if ##yRz## holds, then ##y > z##. So person ##x## is taller than person ##y## and person ##y## is taller than person ##z##.

It must be the case that person ##x## must be taller than person ##z##.

Yes, i agree with what you said...but the question does not mention that y > z

kingstar said:
Yes, i agree with what you said...but the question does not mention that y > z

According to your definition (with some slight corrections):

Recall that a relation ##R## defined on a set ##A## is transitive if ##\forall x, y, z \in A##, if both ##xRy## and ##yRz## hold, then ##xRz## holds as well.

Notice I added the word 'if'. If you assume they hold, does the conclusion still hold?

Yeah then it holds. The definition is provided as part of the information the exam question but i'll just assume it in the exam as well. Thanks

1. What is a transitive set?

A transitive set is a mathematical concept that refers to a set in which every element is also a subset of the set. This means that if x is a member of a transitive set, then every element of x is also a member of the set.

2. How is a transitive set different from other sets?

Unlike other sets, a transitive set follows the property of transitivity, which states that if x is an element of a set, then all the elements of x are also elements of the set. This creates a hierarchical structure within the set.

3. What is the importance of transitive sets in mathematics?

Transitive sets are important in mathematics because they allow for the creation of well-ordered sets, which are sets that can be arranged in a specific order. This is useful in many areas of mathematics, such as set theory and topology.

4. Can all sets be considered transitive sets?

No, not all sets can be considered transitive sets. Only sets that follow the property of transitivity can be considered transitive sets. For example, the set {1, 2, 3} is not transitive because 2 is not a subset of the set, while the set {{1, 2}, {3, 4}} is transitive because every element of the set is also a subset.

5. How are transitive sets used in computer science?

In computer science, transitive sets are used in the creation of data structures such as trees and graphs. They are also used in algorithms that require a hierarchical structure, such as sorting and searching algorithms.

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