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The relation 'has the same father as' over the set of children is:

The relation 'has the same father as' over the set of children is: Correct Answer an equivalence relation

Concept:

Reflexive relation:

A relation R is said to be reflexive if for an element a aRa.

Symmetric relation:

A relation R is said to be symmetric if for an element a and b we have aRb implies bRa.

Transitive relation:

A relation R is said to be transitive if for elements a, b and c we have aRb, bRc then aRc.

Equivalence relation:

A relation R is said to be an equivalence relation if it is reflexive, symmetric and transitive.

 

Calculation:

For a, b in the set of children we say that a and b are related if a has the same father as b.

Now we will check for each property one by one.

The relation is obviously reflexive as a has the same father as a.

Therefore, aRa.          .... (1)

If a has the same father as b then that means both a and b have the same father.

Therefore, b also has the same father as a. Hence, the relation is symmetric.

Therefore, aRb implies bRa.         .... (2)

Now, assume that aRb and bRc.

That means a has the same father as b and b has the same father as c. This implies that a, b and c have the same father.

Therefore, a has the same father as c. Thus, the relation is transitive.

Therefore, aRb, bRc implies aRc.        .... (3)

Thus, from (1), (2) and (3) we conclude that the given relation is an equivalence relation.

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Fact 2: Two of the children have blue eyes and two of the children have brown eyes.
Fact 3: Half of the children are girls.

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II: Two of the children are boys.
III: The boys have brown eyes.