10/10/19
Lecture 2 – Complex Sets
All sets have cardinality, which is the number of elements in the set. It is given as the modulus of the
set. For Example:
A = {1,2,5,4}
|A| = 4
A Singleton is a type of set with only one element and so has a cardinality of 1.
Sets can also be made up of other sets, each of which only counts for one element in the set. For
example:
A ={1, {1, 2}, {1, 2, 3}, 0}
|A| = 4
Types of Set
Empty Sets: Also called a Void or Null Set, is an empty set, is a set with no members, and so a
cardinality of 0. It is given by the symbol: Ø
Disjoint Sets: Two sets who share no common elements between them. Any set is disjoint with an
empty set. For example:
A = {1,3,5} and B = {2,4,6} are disjoint.
Equal Sets: Two sets which are the same as each other. They have the same elements although these
can be in different orders. Shown by the symbol: ‘=’ or ‘⊆’ and ‘⊇’
‘A = B’ or ‘A ⊆ B and A ⊇ B’
Not Equal Sets: Where two sets are not equal. This may be as simple as there being one element
different between the two. Shown by the symbol: ‘≠’
Where A = {1, 2, 3} and B = {1, 2}
A≠B
Proper and Improper Subsets and Supersets
Subsets are sets that exist within other sets and supersets are sets that contain other sets.
A = {1, 2, 3}
B = {1, 2, 3, 4, 5} 2
1 B
In this case, A is a subset of B,
A 3
and B is a superset of A, written as: 4
A ⊂ B and B ⊃ A
5
‘Subset Of’ symbol: ⊂
‘Superset Of’ symbol: ⊃
These are examples of Proper Subsets
and Supersets, where A ≠ B.
Lecture 2 – Complex Sets
All sets have cardinality, which is the number of elements in the set. It is given as the modulus of the
set. For Example:
A = {1,2,5,4}
|A| = 4
A Singleton is a type of set with only one element and so has a cardinality of 1.
Sets can also be made up of other sets, each of which only counts for one element in the set. For
example:
A ={1, {1, 2}, {1, 2, 3}, 0}
|A| = 4
Types of Set
Empty Sets: Also called a Void or Null Set, is an empty set, is a set with no members, and so a
cardinality of 0. It is given by the symbol: Ø
Disjoint Sets: Two sets who share no common elements between them. Any set is disjoint with an
empty set. For example:
A = {1,3,5} and B = {2,4,6} are disjoint.
Equal Sets: Two sets which are the same as each other. They have the same elements although these
can be in different orders. Shown by the symbol: ‘=’ or ‘⊆’ and ‘⊇’
‘A = B’ or ‘A ⊆ B and A ⊇ B’
Not Equal Sets: Where two sets are not equal. This may be as simple as there being one element
different between the two. Shown by the symbol: ‘≠’
Where A = {1, 2, 3} and B = {1, 2}
A≠B
Proper and Improper Subsets and Supersets
Subsets are sets that exist within other sets and supersets are sets that contain other sets.
A = {1, 2, 3}
B = {1, 2, 3, 4, 5} 2
1 B
In this case, A is a subset of B,
A 3
and B is a superset of A, written as: 4
A ⊂ B and B ⊃ A
5
‘Subset Of’ symbol: ⊂
‘Superset Of’ symbol: ⊃
These are examples of Proper Subsets
and Supersets, where A ≠ B.
