Let A be the set of people on Oahu, say for definiteness at 12 noon on October 1

Question

Let A be the set of people on Oahu, say for definiteness at 12 noon on October 19th, 2016. Let R be the relation on A defined by aRb is b is at least as old as a. Let S be the relation on A defined by aSb is a and b were born in the same year.

(a) One of R and S is an equivalence relation, one is a partial order. Which is which (no justification needed)?

(b) For whichever of S and R is a partial order, does it have a minimal element? Could it have more than one minimal element? Explain your answers.

(c) For whichever of S and R is an equivalence relation, estimate the number of equivalence classes. Explain your answer.

Explanation / Answer

a) R is a partial order. S is an equivalence relation.

b) R is a partial order. A minimal element of R, say m, is an element such that mRa for all a. This means a is at least as old as m for all a. This m can be taken to be the youngest living person in Oahu as on 12 noon, Oct 19, 2016. Since m is the youngest person, nobody can be younger than him/her. Thus, mRa is always true and m becomes a minimum element.

Such a minimum element need not be unique. It is possible for 2 people to be born at exactly the same time (down to the second). In this case, we can have multiple minimal elements which are the youngest people in Oahu,

c) Two people are related in S if tey are born in the same year. The equivalence classes thus consists of the set of people born in 2016, 2015, 2014 .... .

Theoretically, there is no definite limit to this, but on a particular date, at a particular time, there must be an oldest person living in Oahu. Let's say he/she/they were born in year Y, Then, the distinct equivalence classes are [Y], [Y+1], [Y+2] ... [2016]

Then, the number of equivalence classes is the number of years from year Y to the year 2016. This equals 2016-Y+1

That is, 2016-Y+1 which is also one more than the age of the person (2016-Y)

As we can safely assume that the age of the oldest person in Oahu is not more than 150 (roughly), the number of distinct equivalence classes is less than 151.

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