Question

Let X be the set of all persons living in a city. Persons x, y in X are said to be related as x < y if y is at least 5 years older than x. Which one of the following is correct?
A. The relation is an equivalence relation on X
B. The relation is transitive but neither reflexive nor symmetric
C. The relation is reflexive but neither nor symmetric
D. The relation is symmetric but neither transitive nor reflexive

Answer 1

Hint: In order to solve this problem we need to check the relation that if it is reflexive, symmetric or transitive with respect to their definition. In mathematics, a homogeneous relation R over a set X is transitive if for all elements a, b, c in X, whenever R relates a to b and b to c, then R also relates a to c. Knowing this will solve your problem and will give you the right answer.

Complete step-by-step answer:
In relation and functions, a reflexive relation is the one in which every element maps to itself.
It is given that X is the set of all persons living in a city.
Then the Relation(R) can be {(x, y): x, y ∈ R & y is 5 years older than x}

For Reflexive:
In relation and functions, a reflexive relation is the one in which every element maps to itself.
Here we will take both variables as same so, we do
(x, x) should $ \in $ R for all x $ \in $ X
Now, x cannot be 5 years older than itself clearly. So the relation is not reflexive.

For Symmetric:
A symmetric relation is a type of binary relation. An example is the relation "is equal to", because if a = b is true then b = a is also true. Formally, a binary relation R over a set X is symmetric.
Here we have to relate x and y with y and x.
If (x, y) $ \in $ R $ \Rightarrow $ (y, x) $ \in $ R
(x, y) $ \in $ R $ \Rightarrow $ y is at least 5 years older than x.
(y, x) $ \in $ R $ \Rightarrow $ x is at least 5 years older than y. This contradicts the above statement. Hence the relation is not symmetric.

For Transitive:
A homogeneous relation R over a set X is transitive if for all elements a, b, c in X, whenever R relates a to b and b to c, then R also relates a to c.
Here we have to consider a third variable and relate it as per the definition.
If (x, y) $ \in $ R and (y, z) $ \in $ R $ \Rightarrow $ (x, z)∈R.
(x, y) $ \in $ R $ \Rightarrow $ y is at least 5 years older than x.
(y, z) $ \in $ R $ \Rightarrow $ z is at least 5 years older than y.
Then, (x, z) $ \in $ R $ \Rightarrow $ z is at least 5 years older than x.
Since, z is at least 10 years older than x. The relation is transitive.

Hence, the relation is transitive.
So the correct option is B.

Note: When you get to solve such problems you need to recall the definitions of various relations and then we have to check the given relation is such that it is reflexive, symmetric and transitive. One of the tricks to recall the definitions of each reflexive, symmetric and transitive relation is that reflexive includes a single variable, symmetric includes two variables and transitive includes three variables. Knowing this will help you remember the various definitions.