Determining whether a relation is transitive or not.

Question

While trying to determine whether the following relations are transitive or not, I got stuck in between. The following are the two relations -

1. Relation R in the set $\mathbb{N}$ of natural numbers defined as $$R = \{(x,y): y = x + 5 \ \text{and} \ x < 4 \} $$

I know that domain of the above relation will be : $\{1,2,3\}$ and the range of the above relation will be : $\{6,7,8\}$ .

2. Relation R in the set A of human beings in a town at a particular time given by - $$ R = \{(x,y) : x \ \text{is wife of} \ y\} $$

For this, I know that domain of the above relation will be : $\{\text{married girls}\}$ and the range of the relation will be : $\{\text{married males}\}$ .

From the definition of a Transitive Relation -

"In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c. " - Transitive Relation Definition(Wiki)

For the 1st Case :

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

I don't think that this obeys the definition of transitive relation.

For the 2nd Case :

$$R = \{({MG}_1,{MM}_1),({MG}_2,{MM}_2), \dots\} $$ Where: ${MG}_1$ = Married Girl(1); ${MM}_1$ = Married Male(1)

Again, if it had been a transitive relation, then there should be elements - $$\{({MG}_1,{MM}_1),({MM}_1,{MG}_2),({MG}_2,{MG}_1)\}$$

Now, obviously, a wife can not be the wife of other girl (wlog).

So, it should not be a transitive relation too. But, according to the marking scheme, it is a transitive relation. Am I getting something wrong? Any help will be greatly appreciated.

NOTE : This is not a homework/assignment/test question. It's a past year question which I encountered with while solving some model papers for proper preparation of my Exam. (For future reference, if any concerned.)

Answer 1

Remember that the implication:

$$p \rightarrow q$$

Is always true when $p$ is false, regardless of the truth value of $q$.

Thus, consider your first relation $R = \{(1,6),(2,7),(3,8)\}$. Note that: $$(x,y)\in R \wedge (y,z)\in R$$

is always false, hence the implication:

$$(x,y)\in R \wedge (y,z)\in R \rightarrow (x,z)\in R$$

is always true, and hence $R$ is transitive.

By the same principle, your second relation is also transitive.

Answer 2

Both relations are transitive by vacuity. In the first case you would need to prove that, for example, if 1 is related to 6 and 6 is related to some $c$ then $1$ is related $c$. Since 6 is not related to anything the condition for transitivity holds.

In the same way, for the second case you would need to proe that: if $x$ is wife of $y$ and $y$ is wife of $z$, then $x$ is wife of $z$, again this is satisfied because $y$ is never wife of anyone since it is a men.