Relation between symmetric and reflexive relations | Relation | Discrete Mathematics

In mathematics, relations are used to describe the relationship between two objects or sets. Symmetric and reflexive relations are two important types of relations that we encounter frequently in discrete mathematics.
A relation R on a set A is said to be symmetric if for any a,b in A, (a,b) ∈ R implies (b,a) ∈ R. In other words, if a is related to b, then b is also related to a. For example, the relation “is equal to” is a symmetric relation as a=b implies b=a.
On the other hand, a relation R on a set A is said to be reflexive if for any a in A, (a,a) ∈ R. In other words, every element in the set A is related to itself. For example, the relation “is greater than or equal to” is a reflexive relation as a ≥ a for any a.

Now, let’s look at the relationship between these two types of relations. It turns out that if R is both symmetric and reflexive, then R is also transitive. Transitivity means that if a is related to b and b is related to c, then a is also related to c.

To prove that a relation R is transitive, we can use the fact that R is reflexive to show that a is related to a, and then use the fact that R is symmetric to show that if a is related to b, then b is related to a. Finally, we can use the fact that R is symmetric again to show that if b is related to c, then c is related to b. Using these three facts, we can then conclude that a is related to c, which shows that R is transitive.

In summary, symmetric and reflexive relations are important concepts in discrete mathematics, and their relationship to transitivity is an interesting and useful result. We hope that you found this video informative and helpful in understanding these concepts better. Thank you for watching!
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