# 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.