In the question below determine whether the binary relation is: (1) reflexive, (2) symmetric, (3) antisymmetric, (4) transitive.

a) the relation r on the set of all people where arb means that a is younger than b.

The relation is antisymmetric and transitive

Step-by-step explanation:

Let a,b,c be elements of the set of all people.

1) Let a be a person who is 20 years old. aRa means that this person is younger than themselves, which it's false because 20<20 is false. Then R is not reflexive.

2) Let a be a person who is 20 years old and b a person who is 30 years old. Then a is younger than b, that is, aRb.

However, it is not true that b is younger than a, as 30<20 is false, therefore bRa is false and R is not symmetric.

3) Suppose that aRb, so that a is younger than b. Then, b is not younger than a. If n denotes the age of a and m denotes the age of b, we have that n<m which implies that m<n is false. Then bRa is false, thus R is antisymmetric.

4) Suppose that aRb and bRc. Let n,m,p denote the ages of a,b,c respectively. Then n<m and m<p (a is younger than b and b is younger than c), and by transitivity of the ordering of numbers, n<p, that is, a is younger than c. Thus aRc, and R is transitive.

step-by-step explanation:

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