Set Theory

Set

A set is a way to express a collection of objects.

Contains object and order is not important. Sets are written as a series of elements separated by commas and enclosed in curly brackets.

cardinality of a set :: |A|

The cardinality of a set refers to the number of items in a set and it's denoted using the symbol that you normally see with absolute value.

It is defined only for finite sets.

Equal Set

When two sets have exactly the same elements, they are considered equal.

Equivalent Set

If the cardinalities of two sets are same, they are called equivalent sets.

Empty Set or Null Set

When a set has no elements, it's called an empty set.

Subset : example Venn diagram

This diagram represents pictorially how these special sets relate to each other. You'll notice that N is the smallest set. And N is a subset of W. W is a subset of Z. Z is a subset of Q. And finally, Q is a subset of all real numbers. Some other types of sets.term-9

Singleton Set or Unit Set

Singleton set or unit set contains only one element. A singleton set is denoted by {s}.

is an element of :: ∈

To indicate that an object is an element of a set, we use this symbol, which looks a little bit like E.

We can also say that the letter f is not an element of the set A. This is denoted by putting a line through the element of symbol.

subset :: ⊆

If every element in set A is also an element of set B, then A is considered a subset of set B, written as A subset B.

Proper Subset :: ⊂

"subset of but not equal to"

If A is a subset of B and there is an element of B that is not in A, then A is considered a proper subset of B.

Universal Set :: U :: Universe of Discourse

It is a collection of all elements in a particular context or application. All the sets in that context or application are essentially subsets of this universal set. Universal sets are represented as U.

Set Intersection

It's written as A intersects B and the formal notation says that the result of the intersection of sets A and B is the set of all values X such that X is an element of A, and X is an element of B.

Set Union

is A union B. And this definition says the set of all values X, where X is an element of A, or X is an element of B, or both.

Set Difference

which takes the elements in set A minus the elements in set B. The formal definition says the set of all values X such that X is an element of A, and X is not an element of B.

A−B={x|x∈A AND x∉B}

Set Complement :: A′={x|x∉A}

Set Complement is written as A with a line over it, or A raised to the C power.The formal definition says that A complement is the set of all values X such that X is not in the set A. So it's anything, except for the values that were in set A.

power sets

A power set is the set of all subsets for a particular set including the empty set. The cardinality of a power set is two to the n, where n is the cardinality of the original set. For example, if the cardinality of set A is equal to three, which means that A has three elements, then the cardinality of the power set of A, which is denoted with a script P in front of the A, is equal to eight, because two-cubed is equal to eight.

Disjoint

Two sets are disjoint if they have no elements in common. In other words, if the intersection of the sets is the empty set.

set partition

A set partition includes all disjoint subsets of a set that when combined make it whole.

Notice, none of subsets have anything in common. They are all disjoint, but when put together they make up the whole.

Cartesian products

Cartesian products are a result of taking two sets, such as sets A and B, and creating a list of ordered pairs. A more formal definition looks like this. A cross B is equal to the ordered pairs of a comma b, such that little A is an element of set A, and little B is an element of set B.

To predetermine the number of ordered pairs we can take the cardinality of set A times the cardinality of set B. That gives us the number of ordered pairs.

two ordered n-tuples are equal if and only if each corresponding pair of elements is equal.

Identity Law

A∪∅=A, A∩U=A

Idempotent Law

A∪A=A, A∩A=A

Domination Law

A∪U=U , A∩∅=∅

Complementation Law

(A')'=A

Communtative Laws

A∪B=B∪A , A∩B=B∩A

Associative Law

A∪(B∪C)=(A∪B)∪C , A∩(B∩C)=(A∩B)∩C

Distributive Law

A∩(B∪C)=(A∩B)∪(A∩C) , A∪(B∩C)=(A∪B)∩(A∪C)

Absorption Law

A∪(A∩B)=A and A∩(A∪B)=A

De Morgan's Law

(A∩B)' = A'∪B' , (A∪B)'=A'∩B'

Complement Law

A∪A'=U, A∩A'=∅

Difference Equivalence

A-B=A∩B'

|AUB| =

|A| + |B| - |A∩B|

|A-B|

|A| - |A∩B|