Sets and Venn Diagrams
Sets and Venn Diagrams
Sets Basics
- A set is a collection of distinct objects, termed as elements.
- The elements of a set can be specified in a listing enclosed by braces, for example, the set of natural numbers less than 5 is denoted as {1, 2, 3, 4}.
- The Universal set (𝕌) is the set that contains all the elements under consideration for a particular discussion or problem.
- A subset (⊆) of a set is a set containing elements that all belong to the original set. For example, if set A = {1,2,3}, then A, {1}, {2}, {3}, {1,2}, {1,3}, {2,3} and the empty set {} are subsets of A.
Venn Diagrams Basics
- A Venn diagram is a pictorial depiction of sets by means of overlapping circles, where each circle represents a set.
- The universal set is generally represented by a rectangle, and subsets of the universal set are represented as circles within this rectangle.
- Overlapping regions in the Venn diagram represent common elements of the sets.
Intersection and Union of Sets
- The intersection of two or more sets, denoted as A ∩ B, includes all the elements that are common to all the sets.
- The union of two or more sets, denoted as A ∪ B, includes all the elements that are in any of the sets.
- For example, if A = {1, 2, 3} and B = {3, 4, 5} then A ∩ B = {3} and A ∪ B = {1, 2, 3, 4, 5}.
Relative Complement and Absolute Complement
- Relative complement or difference of sets is the set of elements that belong to one set but not the other. It’s denoted as A - B or A \ B.
- The absolute complement of a set A, denoted by A’ or A^c, is the set of elements which are in the universal set but not in A.
De-Morgan’s Laws for Sets
- De-Morgan’s Laws are used for manipulating expressions in set theory.
- The first law says the complement of the union of two sets is equal to the intersection of their complements. Symbolically, (A ∪ B)’ = A’ ∩ B’.
- The second law says the complement of the intersection of two sets is equal to the union of their complements. Symbolically, (A ∩ B)’ = A’ ∪ B’.
Conditional Probability and Independent Events
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Conditional probability is the probability of an event given the occurrence of another event. It is denoted as **P(A B)** which reads as the probability of A given B. If P(B) ≠ 0, then **P(A B) = P(A ∩ B) / P(B)**. - If the occurrence of event A does not change the probability of occurrence of event B, the events A and B are said to be independent such that P(A ∩ B) = P(A)P(B).
Venn Diagrams with Three Sets
- A Venn diagram can include three sets, typically represented by three overlapping circles.
- Here, the intersection of all three sets A ∩ B ∩ C is the region where all three circles overlap.
- There are seven distinct regions formed by the three circles. So, a universal set with three subsets can have 2^3-1=7 types of pairwise disjoint non-empty subsets.