Sets and Sample Spaces

• Sets and sample spaces form the fundamental elements of mathematical language and symbols.

Definition of Sets and Sample Space

• A set comprises distinct objects, termed elements, enclosed in curly braces {}.
• For example, A={1,2,3,4} is a set.
• Sample space (S) in probability refers to all possible outcomes of an experiment.
• For example, flipping a coin yields a sample space S={H,T}, where H represents heads and T represents tails.

Types of Sets

• A finite set contains a countable number of elements, such as B={2,4,6,8}.
• An infinite set contains an uncountable number of elements, such as C={1,2,3,...}.
• An empty or null set (∅) contains no elements, such as the set of prime numbers divisible by 4.
• A universal set (U) contains all objects under consideration.
• A subset (⊆) is a set whose elements are all contained within another set, such as X={1,2}⊆Y={1,2,3,4}.

Set Notation and Symbols

• The notation a ∈ A indicates that "a is an element of A".
• The notation 3 ∈ {1,2,3,4,5} signifies that 3 is an element of the set {1,2,3,4,5}.
• The notation x ∉ {a, b, c} means x is not an element of the set {a, b, c}.
• A ⊆ B denotes that A is a subset of B, as in {1,2} ⊆ {1,2,3,4}.
• A ⊂ B signifies that A is a proper subset of B, meaning A is a subset of B but not equal to B, for example {a, b} ⊂ {a, b, c}.
• |A| represents the cardinality of set A, indicating the number of elements in A.
• For instance, if A={a, b, c, d}, then |A|=4.

Set Operations

• The union of sets A and B, denoted A∪B, includes all elements in A, in B, or in both.
• A∪B= {x|x∈A OR x∈B}.
• For example, if A = {1,2,3} and B= {3,4,5}, then A ∪ B = {1,2,3,4,5}.
• The intersection of sets A and B, denoted A∩B, includes elements present in both A and B, expressed as A∩B= {x|x∈A AND x∈B}.
• For example, if A = {2, 5, 7} and B = {1, 2, 5, 8}, then A∩B = {2, 5}.
• The complement of set A (A′) includes elements not in A, where A′={x|x∉A}; specifically, A′=(U−A), with U being the universal set.
• If U= {1,2,3,4,5,6} and A= {1,2,3}, then A′= {4,5,6}.
• Set difference (A–B) includes elements only in A but not in B, expressed as A−B= {x|x∈A AND x∉B}.
• If A = {1,2,3,4} and B = {3,4,5,6}, then A – B = {1,2}.
• The power set P(A) represents the set of all subsets of A.
• If A={1,2}, then P(A)={∅,{1},{2},{1,2}}.

Sample Space in Probability

• The sample space is the total set of all possible outcomes.
• Rolling a six-sided die gives a sample space of S={1,2,3,4,5,6}.