Discrete Math - Naïve Set Theory

A set is a well-defined collection of objects. You can represent sets using tabular form, for instance N={1,2,3,…}. You can represent sets using set builder form, for instance N={x|x=natural numbers}. In a set order does not matter. 

A null set is a set with no elements in it. It can be represented as { }  or ∅. A proper subset, for instance A⊆B, states that all elements in A are in B. Example: {1,2}⊆{1,2,3}. However, one rule should be remembered, the empty set is not a subset of the set of the empty set. This is a general rule, {a}⊄{{a}}. A set with all elements in it (relative to context is the Universal set denoted U, aka domain of discourse).

A union of sets is denoted by A∪B, which denotes all elements in sets A or B. In set builder notation, we can say: A∪B={x|x∈A∨x∈B}. There are a couple of laws that follow:

1. Commutative Law: A∪B=B∪A
2. Associative Law: (A∪B)∪C=A∪(B∪C)
3. Idempotent Law: A∪A=A
4. A∪U=U iff A⊂U
5. A∪∅=A

An intersection of sets is denoted by A∩B, which denotes all elements in sets A and B. In set builder notation, we can say: A∩B={x|x∈A∧x∈B}. There are a couple of laws that follow:

1. Commutative Law: A∩B=B∩A
2. Associative Law: (A∩B)∩C=A∩(B∩C)
3. Idempotent Law: A∩A=A
4. A∩U=A iff A⊂U
5. A∩∅=∅

The distributive law is the following:

1. A∩(B∪C)=(A∩B)∪(A∩C)
2. A∪(B∩C)=(A∪B)∩(A∪C)

We can see this in Venn diagrams:

Disjoint sets are two sets where there are no elements in common i.e.: A∩B=∅. Differences of sets are denoted as A-B or A\B, this is simply the set of elements in A but not B. This means A-B={x|x∉B∧x∈A}.

A compliment of a set is the difference between the universe and it’s subset i.e.: $ A^c=U-A $. This means that $ A^c={x|x∈U∧x∉A} $. Compliments allow for the following rules:

1. $ A∪A^C=U $
2. $ A∩A^C=∅ $
3. $ (A^C )^C=A,∅^c=U,U^c=∅ $
4. DeMorgans Laws: $ (A∪B)^C=A^C∩B^C, (A∩B)^C=A^C∪B^C $

The following sets are important to look at (types of numbers):

Table 17