A∩B Formula

A Intersection B Formula is an important mathematical formula used to find the common region between two sets A and B. A intersection B formula in mathematical notation is represented with the Intersection Symbol “∩” i.e., A∩B. The Intersection symbol when used with different sets is used to denote the intersection of those selected sets. This can be represented as (A ∩ B) represents that both set A and set B occur at the same time. In other words, (A ∩ B) represents elements from both set A and set B.

In this article, we will learn A intersection B Formula including the Venn diagram for A intersection B and the intersection of three sets A, B, and C in detail to understand the concept better along with some solved examples and practice problems.

What is A Intersection B?

(A ∩ B) read as A intersection B represents that both set A and B occur at the same time. A intersection B represents the common region or common elements between two sets.

When we say A ∩ B, we’re referring to the set where both A and B occur. In simple terms, it’s all the things that are in both A and B at the same time.

Property of (A∩B)

A intersection B follows the commutative law. Hence, the commutative law for A intersection B is given as

(A∩B) = (B∩A)

A Intersection B Formula

Let’s say we have two sets, A and B, within a universal set U. Then the formula for number of elements in A intersection B is given as below:

n(A∩B) = n(A) + n(B) – n(A U B)

Where,

• n(A): This is the number of elements that belong to set A, and
• n(B): This is the number of elements that belong to set B, and
• n(A U B): This is the number of elements in total of set A and B.

It’s essentially saying that you can find the size of the intersection by adding the sizes of A and B and then subtracting the size of their union (A U B).

A Intersection B Venn Diagram

We know that A intersection B is the common region between two sets A and B. This common region between set A and Set B is represented using Venn Diagram added below:

Probability of A Intersection B

Probability of A intersection B denotes the chances of occurrence of both set A and B at the same time out of all the possible in the universal set. We can calculate the probability of both A and B happening simultaneously. Now, to find the probability of set A (P(A)), we use the formula:

P(A) = (Number of elements in A set) / (Total number of possible elements in set)

Similarly, for set B the probability P(B) is given as

P(B) = (Number of elements in B set) / (Total number of possible elements in set)

To find the probability of both set A and B happening together P(A ⋂ B)

P(A ⋂ B) = (Number of elements in A ⋂ B) / (Total number of possible elements in set)

If the given sets A and B are independent sets and are also independant of each other. Then we can say that the probability of both sets A and B are also independant of each other it signifies that any change in occurrence of Probability of set A will not affect occurrence of Probability of set B.

Probability formula for independant sets can be rewritten as:

P(A∩B) = P(A) × P(B)

Also, to note here that if A and B are mutually exclusive sets i.e have nothing in common then the the probability of both sets A and B is also mutually exclusive.

Probability formula for mutually exclusive sets can be rewritten as:

P(A⋂B) = 0

A Intersection B Intersection C

(A∩B∩C) read as A intersection B intersection C represents the common elements between three sets namely, set A, B and C occurring at the same time. This means we’re looking at both sets happening together. The specific formula used depends on whether these sets are connected or not.

When we say A ∩ B ∩ C, we’re referring to the set where all three A, B and C occur. In simple terms, it’s all the things that are in all three A, B and C at the same time.

A Intersection B Intersection C Properties

A intersection B intersection C follows the property of associativity. Hence, the associative law for A intersection B intersection C is given as

(A∩B)∩C = A∩(B∩C)

A Intersection B Intersection C Formula

Let’s say we have three sets: A, B and C within a Universal Set S. Universal Set ‘S’ is the collection of all possible elements.

The formula is given as below:

n(A ∩ B ∩ C) = n(A) + n(B) + n(C) – n(A ∪ B) – n(B ∪ C) – n(C ∪ A) + n(A ∪ B ∪ C)

where,

• n(A) is the number of elements that belong to set A.
• n(B) is the number of elements that belong to set B.
• n(C) is the number of elements that belong to set C.
• n(A ⋂ B) is the number of elements that belong to both set A and set B.
• n(B ⋂ C) is the number of elements that belong to both set B and set C.
• n(A ⋂ C) is the number of elements that belong to both set A and set C.
• n(A U B U C) is the number of elements in total of set A, set B and set C.
• n(A ⋂ B ⋂ C) is the number of elements that belong to all the three set A, set B and set C.

The above formula calculates the number of elements in the intersection of A, B and C. It’s essentially saying that you can find the intersection of A, B and C by finding the common intersection of ‘set A intersection set B’ with ‘set B intersection set C’.

A intersection B intersection C Venn Diagram

The venn diagram of A intersection B intersection C represents the are common to the three sets A, B and C. The venn diagram representation for this is represented below:

A Intersection B Union C

A∩(B∪C) represents the intersection of set A with the union of set B with set C. This means we’re looking at the intersection or common elements in set A with (B and C) together taken as a single set. The specific formula used depends on whether these set are connected or not. When we say A ∩ (B ∪ C), we’re referring to the set where intersection of A with set B and C.

A Intersection B Union C Property

A intersection B union C follows the distributive property. Hence, the distributive property rule for A intersection B is given as

A∩(B ∪ C) = (A∩B) ∪ (A∩C)

A Intersection B Union C Formula

Let’s say we have three sets: A, B and C within a Universal Set S. Universal Set ‘S’ is the collection of all possible elements. The formula for number of elements

n(A ∩ (B ∪ C)) = n(A ∩ B) + n(A ∩ C) – n(A ∩ B ∩ C))

Where,

• n(A) is the number of elements that belong to set A.
• n(B) is the number of elements that belong to set B.
• n(C) This is the number of elements that belong to set C.
• n(A ∩ B) is the number of elements common to set A and B
• n(A ∩ C) is the number of elements common to set A and C
• n(A ∩ B ∩ C) is the number of elements common to set A, B and C

Solved Examples on A intersection B

Example 1: Let’s consider two sets, Set X and Set Y. Set X consists of {apples, bananas, oranges, grapes}, and Set Y consists of {bananas, grapes, peaches}. Find the intersection of Set X and Set Y

Solution:

To find the intersection of Set X and Set Y, we look for elements that are common to both sets. In this case, the common elements are “bananas” and “grapes.”

Set X ∩ Set Y = {“bananas”, “grapes”}

Therefore, the intersection of Set X and Set Y is {“bananas”, “grapes”}.

Example 2: Imagine you’re at an ice cream parlour. You have a 1/4 chance of choosing chocolate ice cream, a 1/3 chance of adding choco flakes, and a 1/2 chance of adding whipped cream to your ice cream. What’s the probability that you choose chocolate ice cream and add whipped cream?

Solution:

Probability of choosing chocolate ice cream = P(A) = 1/4

Probability of adding whipped cream = P(B) = 1/2

P(A ∩ B) = P(A) ✕ P(B)

= (1/4) ✕ (1/2)

= 1/8

Example 3: Consider two sets, Set A and Set B. Set A contains {oranges, pears, strawberries, kiwis}, and Set B contains {pears, kiwis, bananas}.

Solution:

To find the intersection of Set A and Set B, we’ll identify elements that are common to both sets. In this case, the common elements are “pears” and “kiwis.” Set A ∩ Set B = {“pears”, “kiwis”} Therefore, the intersection of Set A and Set B is {“pears”, “kiwis”}.

Example 4: Consider you have at a set of pens . You have a 1/8 chance of choosing black pen, a 1/2 chance of choosing blue pen, and a 1/5 chance of choosing designer pen cap. What’s the probability that you choose blue pen with designer pen cap?

Solution:

Probability of choosing blue pen = P(A) = 1/2

Probability of choosing designer pen cap = P(B) = 1/5

Using independence:

P(A ∩ B) = P(A) ✕ P(B)

= (1/2) ✕ (1/5)

= 1/10

Example 5: Let’s consider two sets, Set A and Set B. Set A consists of {red, yellow, orange, black, pink} and Set B consists of {pink, white, golden}. Find A intersection B.

Solution:

To find the intersection of Set A and Set B, we look for elements that are common to both sets. In this case, the common element is pink.

Set A ∩ Set B = {pink}

Therefore, the intersection of Set A and Set B is {pink}.

Practice Problems on A intersection B :

Q1. Consider two sets: Set P and Set Q. Set P contains {oranges, pears, strawberries, kiwis}, while Set Q contains {pears, kiwis, bananas}. Find P intersection Q?

Q2. Consider two sets: Set T and Set S. Set T contains {apples, oranges}, while Set S contains { bananas}. Find T intersection S?

Q3. Consider two sets, Set Alpha and Set Beta. Set Alpha contains {lemons, cherries, watermelons, mangoes}, and Set Beta contains {cherries, watermelons, apples}. Find Alpha intersection Beta?

Q4. Consider three sets: Set P, Set Q and Set R. Set P contains {oranges, pears, strawberries, kiwis}, while Set Q contains {pears, kiwis, bananas} and Set R contains {strawberries, kiwis}. Find P intersection Q intersection R?

Q5. Consider three sets: Set A, Set B and Set C. Set A contains {oranges, pears, strawberries, kiwis}, while Set B contains {pears, kiwis, bananas} and Set C contains {strawberries, kiwis}. Find A intersection B union C?

FAQs on A intersection B

1. What is meaning of A Intersection B?

A intersection B denotes that common region between two sets and A and B

2. What is A Intersection B Formula?

The formula of A intersection B is given for number of elements in A intersection B which is given as n(A∩B) = n(A) + n(B) – n(A U B)

3. What is meaning of A ntersection B ntersection C?

A intersection B intersection C denotes the region or the elements common to all the three sets A, B and C

4. State the formula for A Intersection B Intersection C.

The Formula for number of elements of A intersection B intersection C is given as:

n(A ∩ B ∩ C) = n(A) + n(B) + n(C) – n(A ∪ B) – n(B ∪ C) – n(C ∪ A) + n(A ∪ B ∪ C)

5. What is A Intersection B Union C?

A intersection B Union C denotes the intersection or common elements in set A with (B and C) together taken as a single set happening together.

6. State the formula for A intersection B union C.

The formula for number of elements of A intersection B union C is given as n(A ∩ (B ∪ C)) = n(A ∩ B) + n(A ∩ C) – n(A ∩ B ∩ C))