Changes

It was thought that an informal concept of sets ([[Naive set theory]]) was sufficient, however [[Russell's Paradox]] shows that this can lead to a contradiction if sets are allowed to contain themselves. Modern set theory is more formal, and disallows such paradoxical sets.

It was thought that an informal concept of sets ([[Naive set theory]]) was sufficient, however [[Russell's Paradox]] shows that this can lead to a contradiction if sets are allowed to contain themselves. Modern set theory is more formal, and disallows such paradoxical sets.

== Application ==

== Examples ==

There is the set of unborn children who were [[abortion|aborted]], about which striking conclusions can be drawn.  Given the large and diverse number of elements of this set, it would likely include many who could surpass existing athletic and intellectual achievements.  Indeed, many of the world records and [[Nobel Prize]] achievements recognized today would have been outdone by members of this set.

There is the set of unborn children who were [[abortion|aborted]], about which striking conclusions can be drawn.  Given the large and diverse number of elements of this set, it would likely include many who could surpass existing athletic and intellectual achievements.  Indeed, many of the world records and [[Nobel Prize]] achievements recognized today would have been outdone by members of this set.

Define "A" as the set of all false assertions, and "B" as the set of all the Bible passages. Since there are no [[Counterexamples to the Bible]], the intersection between set A and set B is the [[empty set]].

Define "A" as the set of all false assertions, and "B" as the set of all the Bible passages. Since there are no [[Counterexamples to the Bible]], the intersection between set A and set B is the [[empty set]].

==More Applications -- Combinatorics==

== Application -- Combinatorics==

Sets, and their cardinalities, are used in some very important aspects of combinatorics, that is, the study of combinations.  Let

Sets, and their cardinalities, are used in some very important aspects of combinatorics, that is, the study of combinations.  Let

There is a simple formula for <math>n \choose k</math>.  It is

There is a simple formula for <math>n \choose k</math>.  It is

:<math>{n \choose k} = \frac{n!}{n! (n-k)!}</math>.

:<math>{n \choose k} = \frac{n!}{k! (n-k)!}</math>.

In the case of the set <math>G = \{F\ |\ F \subset \{a,b,c,p,q,r\} \and \|F\|=2\}</math>, the set is

In the case of the set <math>G = \{F\ |\ F \subset \{a,b,c,p,q,r\} \and \|F\|=2\}</math>, the set is