Key Differences Between 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 Made Easy
1. What is the cardinal number of a set and how is it determined?
The cardinal number of a finite set is the total count of unique or distinct elements within that set. It essentially represents the 'size' of the set. For a set named A, its cardinal number is denoted as n(A). To determine it, you simply count each element in the set once. For example, if Set A = {2, 4, 6, 8}, then its cardinal number, n(A), is 4.
2. What is the fundamental difference between the union (∪) and intersection (∩) of sets?
The fundamental difference lies in how they combine elements:
- The union of two sets, denoted as A ∪ B, is a new set that contains all the elements from both set A and set B. An element is included if it is in A, in B, or in both.
- The intersection of two sets, denoted as A ∩ B, is a new set that contains only the elements that are common to both set A and set B. An element is included only if it exists in both sets simultaneously.
3. What is the formula to find the cardinal number of the union of two sets?
The formula to find the cardinal number of the union of two finite sets, A and B, is based on the Principle of Inclusion-Exclusion. It is given by: n(A ∪ B) = n(A) + n(B) - n(A ∩ B). This formula ensures that elements common to both sets (the intersection) are not counted twice.
4. Why is the intersection n(A ∩ B) subtracted when calculating the union n(A ∪ B)?
The intersection n(A ∩ B) is subtracted to correct for double-counting. When you add n(A) and n(B) together, the elements that are present in both sets are counted twice—once as part of set A and again as part of set B. By subtracting the number of common elements, n(A ∩ B), you ensure that every element is counted exactly once, which gives the true size of the union.
5. How does the formula for the union of three sets, n(A ∪ B ∪ C), work?
The formula for the union of three sets expands on the Principle of Inclusion-Exclusion. The formula is: n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(A ∩ C) + n(A ∩ B ∩ C). Here's how it works:
- First, we add the cardinalities of all three sets.
- Next, we subtract the intersections of each pair of sets to remove the double-counted elements.
- However, this process incorrectly subtracts the elements common to all three sets. So, we must add back the intersection of all three sets, n(A ∩ B ∩ C), to get the correct count.
6. Can you explain the union and intersection of sets with a real-world example?
Certainly. Imagine a classroom where:
- Set A = {Students who play Cricket}
- Set B = {Students who play Football}
The union (A ∪ B) would be the set of all students who play at least one of the two sports (Cricket, Football, or both). The intersection (A ∩ B) would be the set of students who play both Cricket and Football. This concept is used in surveys and data analysis to understand overlapping and distinct groups.
7. What is the main difference in calculating the union of disjoint sets versus intersecting sets?
The main difference is the presence of an intersection. Disjoint sets have no elements in common, meaning their intersection is an empty set (A ∩ B = ∅), so n(A ∩ B) = 0. Therefore, the formula for their union simplifies to n(A ∪ B) = n(A) + n(B). For intersecting sets, you must use the full formula, n(A ∪ B) = n(A) + n(B) - n(A ∩ B), to avoid overcounting the shared elements.
8. What do notations like n(A') or n((A ∩ B)') mean in the context of sets?
The apostrophe (') denotes the complement of a set. The complement of a set A, written as A', contains all the elements in the universal set (U) that are not in set A. Therefore:
- n(A') means the number of elements not in set A. It is calculated as n(A') = n(U) - n(A).
- n((A ∩ B)') means the number of elements that are not in the intersection of A and B. It is calculated as n((A ∩ B)') = n(U) - n(A ∩ B).
