Cardinal Number of a Set
The cardinal number of a finite set is the number of distinct elements within the set. In other words, the cardinal number of a set represents the size of a set.
The cardinal number of a set named M, is denoted as n(M). Here, M is the set and n(M) is the number of elements in set M.
A union of sets is when two or more sets are taken together and grouped. If set M and set N are a union, then it is written as M ∪ N.
Disjoint Sets: Disjoint sets are sets that have no elements in common and do not intersect. If M and N are disjoint sets, then it can be mathematically represented as M ∩ N = ∅.
a union b formula
If M and N are finite sets and they are disjoint, then the sum of the cardinal numbers of M and N will be the cardinal number of the union of sets M and N.
n(M ∪ N) = n(M) + n(N)
a intersection b
Intersection of Sets: Two sets intersect when they have one or more common elements. Each coon element is a point of intersection for the two sets.
a intersection b formula
When two sets (M and N) intersect, then the cardinal number of their union can be calculated in two ways:
1. The cardinal number of their union is the sum of their cardinal numbers of the individual sets minus the number of common elements.
n(M ∪ N) = n(M) + n(N) - n(M ∩ N)
2. The cardinal number of their union is given by the sum of their uncommon elements and their common elements.
n (M ∪ N) = n (M – N) + n(N – M) + n(M ∩ N)
If M, N, and C are three finite sets that intersect each-other and are in union, their cardinal number can be represented as n(M ∪ N ∪ C).
Union and Intersection of Three Sets Formula
The cardinal number of the union of three sets is the sum of the cardinal numbers of each individual set and the common elements of all three sets, excluding the common elements of pairs of sets.
n(M ∪ N ∪ C) = n(M) + n(N) + n(C) – n(M ∩ N) – n(N ∩ C) – n(M ∩ C) + n(M ∩ N ∩ C)
Probability Union and Intersection
Probability of Union
The probability for a union of sets depends on the compatibility of the events.
Sum Rule: Two events are said to be incompatible events if they are mutually exclusive and cannot occur simultaneously. The probability of incompatible events is given by the sum of the probabilities of the two events.
P(M ∪ N) = P(M) + P(N)
Compatible events are those events that may occur together and are not mutually exclusive. Their probability can be calculated by taking the sum of probabilities of the events and subtracting the times where they occur together.
P(O ∪ G) = P(O) + P(G) - P(O ∩ G)
Probability of Intersection
Product Rule: If two events M and event N must happen in order for a certain outcome to occur, and if M and N are independent events, then the probabilities can be calculated by multiplying the probabilities of M and N.
P(M ∩ N)=P(M)*P(N)
The probability of two dependent events occurring together is given by: P(M ∩ N)=P(M/N)*P(N)
Example: There are a total of 200 boys in class XII. 120 of them study math, 50 students study science and 30 students study both mathematics and science. Find the number of boys who
n(Total) = 200
n(Math) = 120
n(Science) = 50
n(Math ∩ Science) = 30
(i)Study math but not science
Students who study math but not science is basically the total number of math students minus the number of students who study both science and math.
n(only Math)= n(Math)-n(Both)
(ii)Study science but not math
Students who study science but not math is basically the total number of science students minus the number of students who study both science and math.
n(only Science)= n(Science)-n(Both)
(iii)Study math or science
Students who study science or math is basically the total number of science students plus the total number of math students minus the students who study both science and math.
1. What is a Venn diagram and how to interpret it?
The Venn diagram is a logical representation of all possible relationships between a countable number of separate sets. Venn diagrams represent the elements as points on the plane and sets as the regions enclosed within circles. The dots within the circle show the elements of that group, while the dots are outside the circle show the elements that are not part of a particular set. They are used to arrange elements properly and show the elements that are exclusively present in a particular set, and the elements that are common to two or more sets.
2. Why do we subtract the common elements while calculating the cardinal number of the union of two intersecting sets?
When we add up the individual cardinal numbers of two intersecting sets, the common elements get added twice. They are repeated. Hence, we subtract the common elements so as to ensure that there is no repetition of elements. If this seems too confusing, another way of calculating the cardinal number of the union of two intersecting sets is by considering each section of the Venn diagram separately, that is by adding up the number of unique and uncommon elements, and the number of common elements of the two sets. This again ensures that there is no repetition of the common elements.