Checking relation for reflexive, symmetric and transitive

In summary, the given set of natural numbers implies the relation ##R##, which is not reflexive and symmetric but is transitive. This can be seen using the definition of transitivity and the truth table of implication. For students who have not covered logic and implication, it can be explained that when there are no elements for which a statement might hold, then all elements satisfy the statement.
  • #1
issacnewton
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Homework Statement
Relation ##R## is defined on set ##\mathbb{N}## of natural numbers defined as
$$R = \big \{ (x,y) | y = x + 5 \text{ and } x < 4 \big \} $$
Relevant Equations
Definition of reflexivity, symmetry and transitivity
Now, with the given set of natural numbers, we can deduce the relation ##R## to be as following

$$ R = \big \{ (1,6), (2,7), (3,8) \big \} $$

Now, obviously this is not a reflexive and symmetric. And I can also see that this is transitive relation. We never have ##(a,b) \in R## and ##(b,c) \in R## since if ##(a,b) \in R##, then we have ##b \nless 4## and so we can not have ##(b,c) \in R##. So, in the definition of the transitivity, the antecedent is always false. And since the definition involves an implication, the implication will always be vacuously true. So, the relation is transitive . So, the relation ##R## is transitive but not reflexive or symmetric. Now, I have used the implication and truth table of implication to see that the relation is transitive. I have to teach this to some students and they have not covered logic and implication. They have only studied basics about the methods of proof like, direct proof, proof by contradiction, proof by contra positive etc. So, how would I explain this to these students ?

Thanks
 
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  • #2
If there are no elements for which a statement might hold, then clearly all those elements satisfy the statement (as there are none!).

Not sure what's more to say about it than that. Are you looking for an intuitive answer? Or real life analogies?
 
  • #3
I think your explanation might suffice for these students I will be teaching. Thanks
 

1. What is the purpose of checking for reflexive, symmetric, and transitive relations?

The purpose of checking for these properties is to determine if a relation is well-defined and has certain characteristics that make it useful for various mathematical and scientific applications.

2. How do you check if a relation is reflexive?

A relation is reflexive if every element in the set is related to itself. This can be checked by looking at the ordered pairs in the relation and verifying that each element appears at least once as both the first and second element in a pair.

3. What does it mean for a relation to be symmetric?

A relation is symmetric if for every ordered pair (a,b) in the relation, there is also an ordered pair (b,a) in the relation. In other words, if a is related to b, then b must also be related to a.

4. How can you determine if a relation is transitive?

A relation is transitive if for every ordered pair (a,b) and (b,c) in the relation, there is also an ordered pair (a,c) in the relation. This means that if a is related to b and b is related to c, then a must also be related to c.

5. Can a relation be both symmetric and transitive?

Yes, a relation can be both symmetric and transitive. This means that for every ordered pair (a,b) in the relation, there is also an ordered pair (b,a) in the relation, and if (a,b) and (b,c) are in the relation, then (a,c) must also be in the relation.

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