Lecture 2 - Types of Sets in Set Theory

Empty Set

An Empty Set is a set that contains no objects, which means null or void set.

Notation for the empty set

Notation for the empty set is: {} or symbol ∅

Disjoint Sets

Two or more sets that have no elements in common such as A = {1,2,3,4} and B = {5,7,8,9}.

Non-disjoint sets

Two sets are not disjoint if their intersection is NOT EMPTY.

Equal Sets

Two sets are equal if they share the exact same elements in any order.

Identical elements

They are not equal if they don't have IDENTICAL elements - this means that if one element that is not in the element of the other set, they are not equal, therefore A ≠ B.

Sets of Sets

Sets that can contain atomic elements, which can contain other sets.

Atomic elements

Atomic elements: letters, numbers, pairs of elements etc.

Example of Sets of Sets

Example: A = {a{b,c}} which is a set that contains elements A and another set containing B and C.

Cardinality of Sets

Cardinality: set is the number of its elements.

Cardinality Notation

This is written as |A|.

Example of Cardinality

Example: A = {1,3,4,5,6} then |A| = 5.

Singleton

A set with one element.

Subset

Set A is a SUBSET of Set B if every element of A is also an element of B.

Subset Notation

This is written as A ⊆ B.

Not a Subset

A is NOT A SUBSET of B if there is AT LEAST one element in A that's not in B.

Not a Subset Notation

This is written as A ⊄ B.

Proper Subset

A is considered to be a proper subset of B IF B contains AT LEAST one element that is NOT present in A.

Proper Subset Notation

Symbol for proper subset is: A ⊂ B.

Supersets

B is a superset of A if EVERY element of A is also an element of B.

Superset Notation

This is written as B ⊇ A.

Proper Superset

B is a proper superset of A if there is AT LEAST one element in B which is not in A.

Proper Superset Notation

Written: B ⊃ A.

Universal Set

Nonempty set of all the possible elements, that's relevant to the solution of a specific problem, this does include those of ALL subsets.

Universal Set Notation

Defined using symbol: U.

Complement Sets

Difference between the universe and given set.

Complement Notation

This is denoted by comp(A) = U - A.

Example of Complement Sets

e.g. U = {r,o,y,g,b,I,v} A = {r,y,b} therefore comp(A) = {o,g,I,v}.