Question

Is the relation R consisting of all ordered pairs (a, b) such that a and b are people and have one common parent: reflexive, irreflexive, symmetric, antisymmetric, asymmetric, and/or transitive? If a property doesn’t hold give a counter-example and state the logical definitions of the properties as you consider them.

Answer 1

The relation R is defined by (a,b) so that a and b are people and have one common parent. Then,

1. Reflexive:

Logical Definition: Reflexive property states that for every a in set A (a,a) belongs to R.

So, the condition for relation is a and b must belong to common parent.

Thus, for every (a,a) belongs to R because a and a has common parent.

and similarly, every (b,b) belongs to R.

So, the relation is reflexive.

2. Ir-reflexive:

LOGICAL DEFINITION:The Irreflexive property states that:

For every a in set A, (a,a) not belongs to R.

As per the property, we can say that every a or b belongs to the set. All (a,a) or (b,b) belongs to R as every element is related to itself is true (because two identical elements has same common parent).

because every (a,a) in the set is related.

So, the relation is not Ir-reflexive.

3. Symmetric:

LOGICAL DEFINITION: The property symmetric states that from a set (a,b) belongs to R. If (a,b) belongs to R. Then, (b,a) also belongs to R.

As per the property, the relation is symmetric also because if (a,b) belongs to R. It implies that

(b, a) also belongs to R.

This is because Relation is defined over a and b has common parent.

Hence, the relation is Symmetric.

4. Anti-symmetric:

LOGICAL DEFINITION:A relation is anti-symmetric, if (a,b) belongs to R. It implies that (b,a) not belongs to R unless a=b.

So, as per our relation a and b are related if a and b has common parent.

Then, for every (a,b) belongs to R (because a and b has common parent).

It means that (b,a) also belongs to R.

This contradicts our property.

So, the relation is not Anti-symmetric in nature.

EXAMPLE FOR CONTRADICTION:

If (a,b) belongs to R. Then, a and b has common parent.

Then, (b,a) also belongs to R, for every a and b.

So, our relation is not anti-symmetric.

5. Assymmetric:

LOGICAL DEFINITION:The property of Assymetric property states that: If a and b belongs to a set A. Then, for every (a,b) belongs to R implies that (b,a) not belongs to R. (The difference with anti symmetric and asymmetric is that in asymmetric the condition a=b is not excluded. The condition a=b is also included).

For our relation,

if (a,b) belongs to R. Then, a and b has common parent.

Then, (b,a) also belongs to R because b and a also has common parent.

It contradicts our statement.

So, the relation is not Assymetric.

EXAMPLE FOR CONTRADICTION:

(a,b) belongs to R. Then, a and b has common parent.

It implies that for every (a,b), there will be (b,a) because b and a has common parent.

This contradicts our Assymetric property.

6. Transitive:

LOGICAL DEFINITION:The transitivity property states that if (a,b) belongs to R and (b,c) belongs to R. Then, (a,c) also belongs to R.

As per our relation, if (a,b) belongs to R. It means that a and b has common parent.

and (b,c) belongs to R means that b and c has common parent.

Altogether a, b and c has common parent.

Hence, we can conclude that a and c also has common parent.

It means that (a,c) belongs to R.

Thus, the relation is Transitive.