Is A Relation With One Ordered Pair Transitive?

Yes, a relation containing only one ordered pair is always transitive.

This is because the logical condition required for transitivity is met vacuously. A single ordered pair, such as R={(a,b)}cap R equals the set open paren a comma b close paren end-set

𝑅={(𝑎,𝑏)}

, does not fulfill the premise of the transitive rule, so there is no instance where the rule can be violated.

What is transitivity?

To understand why a single-pair relation is transitive, one must first be clear on the definition of a transitive relation. A binary relation Rcap R

𝑅

on a set Acap A

𝐴

is defined as transitive if, for all elements a,b,a comma b comma

𝑎,𝑏,

and cc

𝑐

in Acap A

𝐴

:

**If (a,b)∈Ropen paren a comma b close paren is an element of cap R

(𝑎,𝑏)∈𝑅

and (b,c)∈Ropen paren b comma c close paren is an element of cap R

(𝑏,𝑐)∈𝑅

, then it must follow that (a,c)∈Ropen paren a comma c close paren is an element of cap R

(𝑎,𝑐)∈𝑅** .

This definition is an "if-then" statement. The key is that the rule only applies if the initial condition—the "if" part—is met.

The case of a single ordered pair

Consider a relation Rcap R

𝑅

that contains only one ordered pair, for example, R={(x,y)}cap R equals the set open paren x comma y close paren end-set

𝑅={(𝑥,𝑦)}

. We must check if this relation satisfies the definition of transitivity by examining if the premise of the rule is ever met.

There are two scenarios to consider:

Scenario 1: The ordered pair is not of the form (a,a)open paren a comma a close paren

(𝑎,𝑎)

Let's use the example R={(1,2)}cap R equals the set open paren 1 comma 2 close paren end-set

𝑅={(1,2)}

.To test for transitivity, we need to find pairs (a,b)open paren a comma b close paren

(𝑎,𝑏)

and (b,c)open paren b comma c close paren

(𝑏,𝑐)

within Rcap R

𝑅

.The only pair is (1,2)open paren 1 comma 2 close paren

(1,2)

. The second element of this pair is 2. Is there any pair in Rcap R

𝑅

that starts with 2? No.Therefore, the condition "**if (a,b)∈Ropen paren a comma b close paren is an element of cap R

(𝑎,𝑏)∈𝑅

and (b,c)∈Ropen paren b comma c close paren is an element of cap R

(𝑏,𝑐)∈𝑅** " is never satisfied. Since the "if" part of the statement is never true, the entire conditional statement is considered true by default in logic. This is known as a vacuously true statement.

In essence, there is no way to construct a counterexample to transitivity, because a counterexample would require finding a sequence of three elements, a→b→ca right arrow b right arrow c

𝑎→𝑏→𝑐

, where the final link a→ca right arrow c

𝑎→𝑐

is missing. A relation with only one pair, like 1→21 right arrow 2

1→2

, simply doesn't have the necessary "chain" to test.

Scenario 2: The ordered pair is of the form (a,a)open paren a comma a close paren

(𝑎,𝑎)

Consider the relation R={(1,1)}cap R equals the set open paren 1 comma 1 close paren end-set

𝑅={(1,1)}

.Here, the elements are a=1,b=1,c=1a equals 1 comma b equals 1 comma c equals 1

𝑎=1,𝑏=1,𝑐=1

. The condition for transitivity requires checking if (1,1)∈Ropen paren 1 comma 1 close paren is an element of cap R

(1,1)∈𝑅

and (1,1)∈Ropen paren 1 comma 1 close paren is an element of cap R

(1,1)∈𝑅

. Since both are true, we must check if (1,1)open paren 1 comma 1 close paren

(1,1)

is also in Rcap R

𝑅

. It is.Therefore, the condition is fulfilled, and the relation is proven to be transitive.

The empty relation: A related example

For further clarity, consider the empty relation, R={}cap R equals the empty set

𝑅={}

, which contains no ordered pairs. The question of its transitivity is resolved in the same way as with a single-pair relation. Because there are no pairs (a,b)open paren a comma b close paren

(𝑎,𝑏)

and (b,c)open paren b comma c close paren

(𝑏,𝑐)

in the relation, the premise of the transitivity rule is never met. As a result, the empty relation is also vacuously transitive.

Conclusion

A relation with only one ordered pair is always transitive, whether the pair is of the form (a,a)open paren a comma a close paren

(𝑎,𝑎)

or not. This is a fundamental concept in discrete mathematics and logic, demonstrating how the truth of an implication depends on the truth of its antecedent. In the single-pair case, the antecedent is not met, so the condition holds without exception.