Set Notation

A set is a collection of distinct elements, also considered an object. Each object in a set is called an element (member) of the set.

Key points about sets:

1. Distinct Elements: Each element in a set is unique. No two elements are the same within a set
2. Order Irrelevance: The order which elements are listed in a set does not matter. For example, the sets {1, 2, 3} and {3, 2, 1} are considered the same
3. Notation: Sets are typically denoted by curly braces. For example, N = {i${}_1$, i${}_2$, …, i${}_k$} could represent a set of players, where each i${}_j$ is a distinct player

Type of Sets §

Empty Set: The empty set, denoted by $\varnothing$ or {}, is a set with no elements

• Example: $\varnothing$ = {} Finite Set: Set with finite number of elements
• Example: {1, 2, 3} Infinite Set: Set with infinite number of elements
• Example: N = {1, 2, 3, …} Subset: Set A is a subset of set B if all elements of A are also elements of B, denoted A $\subseteq$ B
• Example: {1, 2} $\subseteq$ {1, 2, 3} Proper Subset: set A is a subset of set B if A $\subseteq$ B and A $\ne$ B denoted A $\subset$ B
• Example: {1, 2} $\subset$ {1, 2, 3} Universal Set: Set that contains all possible elements for a particular discussion, usually denoted by $\cup$
• Example: N $\cup$ {1, 2, 3} Power Set: The set of all subsets of a set A, including empty set and A itself, denoted P(A)
• Example: A = {1, 2}, then P(A) = {$\varnothing$, {1}, {2}, {1, 2}}

Set Operations §

1. Union: The union of two sets A and B is the set of elements that are in A, in B, or in both, denoted A $\cup$ B
   • Example: {1, 2} $\cup$ {2, 3} = {1, 2, 3}
2. Intersection: The intersection of two sets A and B is the set of elements that are in both A and B, denoted A $\cap$ B
   • Example: {1, 2} $\cap$ {2, 3} = {2}
3. Difference: The difference of two sets A and B is the set of elements that are in A but not in B denoted A - B or A \ B
   • Example: {1, 2} - {2, 3} = {1}
4. Complement: The complement of a set of A is the set of elements that are not in A, usually relative to the universal set $\cup$, Denoted A${}^{{}^{\prime }}$ or A${}^c$
   • Example: if U {1, 2, 3} and A = {1, 2} then A${}^c$ = {3}
5. Cartesian Product: The Cartesian Product of two sets A and B is the set of all ordered pairs (a, b) where a $\in$ A and b $\in$ B denoted A x B
   • Example: if A = {1, 2} and B = {x, y} then A x B = {(1,x),(1,y),(2,x),(2,y)}