Definition formulas properties and solved examples of union and intersection of cardinal number sets
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 b
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)
Union and Intersection of Three Sets
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)
Venn Diagram Union and Intersection Problem Example
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 are 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)
n(only Math)=120-30
n(only Math)=90
(ii)Study science but not math
Students who study science but not math are 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)
n(only Science)=50-30
n(only Science)=20
(iii)Study math or science
Students who study science or math are basically the total number of science students plus the total number of math students minus the students who study both science and math.
n(Science/Math)= n(Science)+n(Math)-n(Both)
n(Science/Math)=50+120-30
n(Science/Math)=140
More About Cardinal Numbers
Cardinal numbers are the numbers that are used for normal counting. They are also called the natural numbers or cardinals. Cardinal numbers set starts from the first digit number 1 and consist of the numbers till infinity. It can be used to describe or represent a quantity. The cardinals do not have values of decimals or fractions. We use these numbers to show the amount of an object or other quantities. Suppose, the question is ‘how many pens do you have?’ then the answer can be 4, 40, 50 etc. thus, all the numbers will come under the category of cardinal numbers. You will learn about cardinal numbers and also the difference between ordinal and cardinal numbers. Vedantu has all the materials you need related to this topic.
FAQs on Union and Intersection of Sets of Cardinal Numbers Explained
1. What is the union of sets of cardinal numbers?
The union of sets of cardinal numbers is the set containing all distinct cardinal numbers that belong to at least one of the given sets. In symbols, for sets A and B, the union is written as A ∪ B.
- If A = {1, 2, 3} and B = {3, 4, 5}, then A ∪ B = {1, 2, 3, 4, 5}.
- Repeated elements are written only once because sets contain unique elements.
- The union combines all elements from both sets.
2. What is the intersection of sets of cardinal numbers?
The intersection of sets of cardinal numbers is the set containing only those cardinal numbers that are common to all given sets. It is written as A ∩ B.
- If A = {1, 2, 3} and B = {2, 3, 4}, then A ∩ B = {2, 3}.
- If there are no common elements, the intersection is the empty set ∅.
- Intersection focuses only on shared elements.
3. What is the formula for the cardinality of union of two sets?
The formula for the cardinality of the union of two sets is n(A ∪ B) = n(A) + n(B) − n(A ∩ B). This formula avoids double counting of common elements.
- n(A) = number of elements in set A
- n(B) = number of elements in set B
- n(A ∩ B) = number of common elements
- Example: If n(A)=5, n(B)=4, n(A ∩ B)=2, then n(A ∪ B)=5+4−2=7.
4. How do you find the union and intersection of two sets step by step?
To find union and intersection, list elements carefully and identify common values first.
- Step 1: Write both sets clearly.
- Step 2: Find common elements → this gives A ∩ B.
- Step 3: Combine all distinct elements → this gives A ∪ B.
- Example: A={2,4,6}, B={4,6,8}
- A ∩ B = {4,6}
- A ∪ B = {2,4,6,8}
5. What is the difference between union and intersection of sets?
The union includes all elements from both sets, while the intersection includes only the common elements.
- A ∪ B → elements in A or B or both.
- A ∩ B → elements in both A and B.
- Example: A={1,2,3}, B={3,4}
- Union = {1,2,3,4}
- Intersection = {3}
6. Can the intersection of two sets be empty?
Yes, the intersection of two sets can be empty if they have no common elements, and it is called the empty set (∅).
- If A={1,2} and B={3,4}, then A ∩ B = ∅.
- Such sets are called disjoint sets.
- For disjoint sets, n(A ∪ B) = n(A) + n(B).
7. What are the properties of union and intersection of sets?
Union and intersection follow standard set properties such as commutative, associative, and distributive laws.
- Commutative: A ∪ B = B ∪ A, A ∩ B = B ∩ A
- Associative: (A ∪ B) ∪ C = A ∪ (B ∪ C)
- Distributive: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
- Identity: A ∪ ∅ = A, A ∩ U = A
8. How do you find the union of three sets?
The union of three sets A, B, and C is the set of all distinct elements in any of the three sets, written as A ∪ B ∪ C.
- List all elements from A, B, and C.
- Remove duplicates.
- Example: A={1,2}, B={2,3}, C={3,4}
- A ∪ B ∪ C = {1,2,3,4}
9. What is the formula for the cardinality of union of three sets?
The cardinality of the union of three sets is given by n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(A ∩ C) + n(A ∩ B ∩ C). This is called the inclusion–exclusion principle.
- Add individual set sizes.
- Subtract pairwise intersections.
- Add the triple intersection.
- This prevents overcounting.
10. Why do we subtract the intersection when finding the union cardinality?
We subtract the intersection because common elements are counted twice when adding n(A) and n(B).
- In n(A) + n(B), elements in A ∩ B appear two times.
- Subtracting n(A ∩ B) removes the extra count.
- This gives the correct value of n(A ∪ B).
- Example: A={1,2,3}, B={3,4}; 3 is counted twice, so subtract 1.
