Regular Expressions

Cardinality

Cardinality refers to the number of elements in a finate set. In the example below, it shows how you would use cardinality.

Example

The set is X

X = {45,62,56,90}

You would say that set X has a cardinality of 4.

A finite set

A finate set is a set that has a limited amount of items.

Infinite sets

Infinte sets, like the name suggests, are sets that have numbers that never end. ℕ,ℚ,ℤ and ℝ are all examples of infinite sets because they don't end.

Countable set

A countable set is either a set that is not infinite or a natural number set(a set with just positive numbers) that is infinite but has numbers which can be counted from one at a time

Example

N = {0, 1, 2, 3, 4 ...}

Set Comprehension

X = {x² | x ∊ ℕ ∧ n < 5}

The X{} means the set "X".

'|' means such that

'∊' means "belong to"

'ℕ' means natural numbers

'∧' means "and"

Altogther, it means,

X eauals x² such tha x belongs to the infinite set of natural numbers, including zero, and n is less than 5.

Remeber that ℕ is the symbol for natural numbers and that it is infinite since there can be an infinite amount of positive numbers.

In this example it means that we have to use the natural numbers of n that is less than 5

ℕ = {0, 1, 2, 3, 4}

and each n is going to be squared

B = {0, 1, 4, 9, 16}

This means that set be equals 0, 1, 4, 9 and 16

Cartesian product of sets

The Cartesian product of two sets is the set of all ordered pairs(a, b), where a is a member of set A and b is a member of set B

Its written as A x B but we say it as "A cross B"

Example

C = A x B

A = {2, 5, 10}

B = {4, 14, 44}

To form the cartesian product, its:

C = {(2, 4), (2, 14), (2, 44), (5, 4), (5, 44), (10, 4), (10, 14), (10, 44)}

Above, we have written each element of set A with each element of set B. To work out how many items in your cartesian product you should have, just do the number of elements in set A times the number of elements in set B.

Subsets and proper sets

In the example below. Every member in set Y is also a member in set X, so we can say Y is a subset of X

We can write this as X ⊆ Y

If X is a subset of Y but is not equal to it, we say that X is a proper set of Y

We write that as X ⊂ Y.

Example

{6,7,3,2,6,8} ⊂ ℕ. This means that the numbers 6, 7, 3, 2, 6, 8 is a proper set of natural numbers. They are not equal to ℕ since ℕ is infinite

Union

By using union, we are able to append all the contents of one set to another

Example

X = {4, 2,6}

Y = {7, 9, 10}

X ∪ Y = {2, 4, 6, 7, 9, 10}

Intersection

Intersection set contains all the comman elements that two sets have

Example

X = {1, 7, 5, 4, 9, 0}

X = {12, 7, 9, 4, 13, 55}

X &cap Y = {4, 7, 9};

Compliment

The difference set consisits elements in one set that are not members of the other set

X = {1, 2, 4, 8, 9}

Y = {1, 8, 5, 10, 12}

X \ Y = {4, 5, 10, 12}

This reads as everything which is in set X but not in set Y.