Introductory concepts

Story in a country store In a nameless backwoods country store, a man enters a store and speaks to the proprietor →

not so serious thinking To unravel the situation, one starts from the confusion of words and try to establish several categories.

Whether you are thinking for this joke or for more serious situation, most thinking follows a similar pattern.

chaos to order You start with chaos or disorder (of lot of things, facts, ideas in no sensible order). Then you classify them as:

these you will not use
these you will use (to establish something new)
combine the new with others to produce the result.

Thus, theory of sets is the mathematical basis for setting things in categories and determining:

which category is part of the larger one
which category overlaps, and
what is excluded.

However, set theory does not just deal with things but also deals where knowledge of a group is important.

WW2 voting debate A debate in United States during the Second World War was to decrease voting age from 21yrs to 18yrs. Thus,

Group:

people (presumably US citizen) in US Army.

Knowledge:

people aged < 21yrs in US Army.

"Importance" of the knowledge is determined by,

Practical Goal:

Change US voting law.

WW2 People We exclude:

Group of people < 18yrs.

We also exclude:

People not in the Army.

Notice that people < 18yrs is a group residing within the group of people not in the Army. Thus, "people < 18yrs are not in the Army".

However, the above statement in not true for "people > 18yrs".

We then exclude:

People in the army but > 21yrs.

Finally, what remains is

People in the army but < 21yrs.

Since, the goal is to change US voting laws, our interest is in the knowledge that soldiers are < 21yrs. Hence, individual differences are not important.

For example, its irrelevant that Jack and Jane are of opposite sexes.