Set Notation

Set Notation

• Sets are collections of unique objects known as elements.
• The notation {a, b, c} signifies a set with elements ‘a’, ‘b’ and ‘c’.
• The order of elements in a set is not significant. {a, b} is the same as {b, a}.

Empty and Universal Sets

• An empty set or null set is a set with no elements, denoted as {} or ∅.
• A universal set contains all elements under consideration, usually denoted by the symbol U.

Membership and Inclusion

• If ‘a’ is an element of a set A, we write a ∈ A. If ‘a’ is not an element of A, we write a ∉ A.
• Inclusion refers to all elements of one set being within another set. If every element in set A is also in set B, we write A ⊆ B.
• A = B signifies that sets A and B are identical: every element of A is in B and every element of B is in A.

Cardinality

• The Cardinality of a set is the number of elements in the set, denoted as

  A

  . For example, if A = {a, b, c}, then

  A

  = 3.

Set Operations

• Union of two sets A and B, denoted as A ∪ B, is a set containing all elements of A and B.
• Intersection of A and B, denoted as A ∩ B, is a set containing common elements of A and B.
• Difference of A from B, denoted as A \ B, is a set of elements in A but not in B.
• Complement of a set A, denoted as A' or A̅, refers to all elements not in A but in the universal set U.

Cartesian Products

• Cartesian Product of two sets A and B, denoted as A × B, is a set of all ordered pairs (a, b) where a belongs to A and b belongs to B. The order of elements does matter in Cartesian product.

Venn Diagrams

• Venn Diagrams are graphical representations of sets and their relations, wherein each set is represented with a circle or oval and elements are represented as points within the shapes. Intersection of sets is visually depicted as overlapping areas.