Sets

Sets

• A set is a collection of well-defined objects.
• The objects in the set are called elements
• Well-defined means it must be clear which objects (elements) are in the set and which are not

Cardinal number

• The cardinal number of a set A is the number of elements in the set A.
• The cardinal number of the set is denoted by or A

Null/empty set

• If a set has no elements, then it is called the null set or the empty set. The cardinal number of the null set is
• The null set is delicted by the symbol or (

Equal sets

• Two sets are equal if they each have the same elements
• For example
    * The sets A = {1, 2, 3, 4} and B = {4, 3, 2, 1} are equal
    * That can be written A = B

Describing a Set

The List Method

• All elements of the set are listed inside curly brackets. For example: {2, 4, 6, 8, 10}
• The order in which you list elements is not important
• An element may appear only once in a set

The Rule Method

• The elements of the set are described using words.
• For example, the set {2, 4, 6, 8, 10} can be written as
    * {Even numbers between 1 and 11}

  * {x|x is an even number between 1 and 11}

Subsets

• If all the elements of a set A are also in a set B, then we say that A is a subset of B
• The symbol for subset is ⊂

Improper subset

• If A is a subset of 8, and A contains all the elements of 8, then A is an improper subset of B

Proper subset

• If A is a subset of 8, but A does not contain all the elements of B, then A is a proper subset of B

Venn Diagrams

• Sets can also be represented using Venn diagrams.

Universal set

• The universal set is the set that contains all elements under consideration
• Every set is a subset of the universal set
• In a Venn diagram, the universal set can be represented by a rectangle
• The letter U is used to denote the universal set

Non-disjoint sets

• If two sets A and B have elements in common, then the sets A and B are represented by overlapping circles.
• The two sets are called non-disjoint sets.

Disjoint sets

• If two sets A and B have no elements in common, they are usually represented by non-overlapping circles.
• If two sets have no elements in common then they are called mutually exclusive sets or disjoint sets.

Union

• The unions of two sets is the set of elements contained in either set
• The union of the two sets A and B is written as A U B
• A U B is the set of elements in A or in B or in both.

Intersection

• The intersection of two sets is the set of elements that are common to both sets.
• The intersection of the two sets A and B is written as A ∩ B
• So, A ∩ B is the set of elements that are in both A and B

Complement of a Set

• The complement of a set A is the set of elements in the universal set U that are not elements of A.
• The complement of a set A is denoted by A' or A∁

Set Difference

• A\B is the set of elements that are in A but not in B.
• A backslash B is pronounced 'A less B'.