CLASS-6
SETS - PROBLEM & SOLUTION

SETS - PROBLEM & SOLUTION -

Example.1) Suppose you have two sets, A and B, where:

Set A represents students who are studying mathematics, and it contains {Alice, Bob, Carol, David}.

Set B represents students who are studying physics, and it contains {Bob, Carol, Eve, Frank}.

Find:

  1. The students who are studying both mathematics and physics.
  2. The students who are studying either mathematics or physics or both.
  3. The students who are studying mathematics but not physics.
  4. The students who are studying physics but not mathematics.
  5. The total number of students studying either mathematics or physics.

Ans.)

  1. The students who are studying both mathematics and physics:-Intersection of sets A and B: A ∩ B = {Bob, Carol}
  2. The students who are studying either mathematics or physics or both:-Union of sets A and B: A ∪ B = {Alice, Bob, Carol, David, Eve, Frank}.
  3. The students who are studying mathematics but not physics:- Set difference of A and B: A - B = {Alice, David}.
  4. The students who are studying physics but not mathematics:Set difference of B and A: B - A = {Eve, Frank}.
  5. The total number of students studying either mathematics or physics:- Cardinality of the union of sets A and B: |A ∪ B| = 6 (There are six unique students in total.)



Example.2) Suppose you have two sets, A and B, and you are asked to find the union and intersection of these sets.

Set A: {1, 2, 3, 4, 5} Set B: {3, 4, 5, 6, 7}

Find:-

  1. The union of sets A and B.
  2. The intersection of sets A and B.

Ans.)

  1. Union of Sets A and B:- The union of two sets, denoted as A ∪ B, contains all the distinct elements that are in either set A or set B or in both.A ∪ B = {1, 2, 3, 4, 5, 6, 7}The union of sets A and B includes all the unique elements from both sets A and B combined.
  2. Intersection of Sets A and B:- The intersection of two sets, denoted as A ∩ B, contains all the distinct elements that are common to both set A and set B. A ∩ B = {3, 4, 5}. The intersection of sets A and B includes only the elements that appear in both sets A and B.



Example.3) Suppose you have two sets, Set A and Set B, with the following elements:

Set A: {1, 2, 3, 4, 5} Set B: {3, 4, 5, 6, 7}

  1. Find the union of Set A and Set B.
  2. Find the intersection of Set A and Set B.
  3. Determine if Set A is a subset of Set B.

Ans.)

  1. Union of Set A and Set B:- The union of two sets, denoted as A ∪ B, contains all the distinct elements that belong to either Set A or Set B or both. A ∪ B = {1, 2, 3, 4, 5, 6, 7}. So, the union of Set A and Set B consists of the elements {1, 2, 3, 4, 5, 6, 7}.
  2. Intersection of Set A and Set B:- The intersection of two sets, denoted as A ∩ B, contains all the distinct elements that are common to both Set A and Set B. A ∩ B = {3, 4, 5}. The intersection of Set A and Set B consists of the elements {3, 4, 5}.
  3. Determining if Set A is a Subset of Set B:- Set A is considered a subset of Set B if every element of Set A is also an element of Set B. This relationship is denoted as A ⊆ B. In this case, Set A is not a subset of Set B because it contains elements (1 and 2) that are not present in Set B. So, A is not a subset of B.



Example.4) Suppose you have two sets, Set A and Set B, defined as follows:

Set A = {1, 2, 3, 4, 5} Set B = {3, 4, 5, 6, 7}

Find the following:

  1. The union of Set A and Set B.
  2. The intersection of Set A and Set B.
  3. The complement of Set A with respect to a universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.
  4. Determine if Set A is a subset of Set B.

Ans.)

  1. Union of Set A and Set B:- The union of two sets, denoted as A ∪ B, contains all the distinct elements from both Set A and Set B. A ∪ B = {1, 2, 3, 4, 5} ∪ {3, 4, 5, 6, 7} = {1, 2, 3, 4, 5, 6, 7}.
  2. Intersection of Set A and Set B:- The intersection of two sets, denoted as A ∩ B, contains only the elements that are common to both Set A and Set B. A ∩ B = {1, 2, 3, 4, 5} ∩ {3, 4, 5, 6, 7} = {3, 4, 5}
  3. Complement of Set A with respect to Universal Set U:- The complement of a set A with respect to a universal set U, denoted as A', contains all the elements in U that are not in A. A' = U - A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} - {1, 2, 3, 4, 5} = {6, 7, 8, 9, 10}.
  4. Determine if Set A is a Subset of Set B:- Set A is considered a subset of Set B if every element of Set A is also an element of Set B. If that's the case, we can say A ⊆ B. A = {1, 2, 3, 4, 5}, B = {3, 4, 5, 6, 7}. All elements of Set A (1, 2, 3, 4, 5) are also present in Set B. Therefore, A is a subset of B, and we can write A ⊆ B.

So, in summary:

  1. A ∪ B = {1, 2, 3, 4, 5, 6, 7}
  2. A ∩ B = {3, 4, 5}
  3. A' = {6, 7, 8, 9, 10}
  4. A ⊆ B (Set A is a subset of Set B).



Example.5) You are given two sets, Set A and Set B, containing integers. Set A consists of numbers from 1 to 10, while Set B contains even numbers from 2 to 12. Find the union, intersection, and complement of Set A and Set B.

Ans.)

Let's solve this problem step by step:

Set A: A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Set B: B = {2, 4, 6, 8, 10, 12}

Now, we'll find the requested operations:

  1. Union (A ∪ B):- The union of two sets contains all unique elements from both sets. A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12}
  2. Intersection (A ∩ B):- The intersection of two sets contains elements that are common to both sets. A ∩ B = {2, 4, 6, 8, 10}
  3. Complement of Set A (A'):- The complement of Set A with respect to a universal set U would contain all elements in U that are not in A. Assuming the universal set U includes all integers from 1 to 12: U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, A' = U - A = {11, 12}
  4. Complement of Set B (B'):- The complement of Set B with respect to the same universal set U: B' = U - B = {1, 3, 5, 7, 9, 11}

So, in summary:-

  • The union of Set A and Set B is {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12}.
  • The intersection of Set A and Set B is {2, 4, 6, 8, 10}.
  • The complement of Set A (A') is {11, 12}.
  • The complement of Set B (B') is {1, 3, 5, 7, 9, 11}.

These operations demonstrate basic set theory concepts involving union, intersection, and complement.



Example.6) You are given two sets, A and B, containing integers. Set A represents the ages of students in a math class, and set B represents the ages of students in a science class. Your task is to find:

a) The number of students who are in both the math and science classes. b) The number of students who are only in the math class. c) The number of students who are only in the science class.

Set A (Math Class):- {16, 17, 18, 19, 20}

Set B (Science Class):- {18, 19, 20, 21, 22}

Find:

a) The number of students in both classes. b) The number of students only in the math class. c) The number of students only in the science class.

Ans.)

a) To find the number of students in both classes, we can calculate the intersection of sets A and B.

Number of students in both classes = |A ∩ B| = |{18, 19, 20}| = 3 students.

b) To find the number of students only in the math class, we need to find the elements that are in set A but not in set B.

Number of students only in math class = |A - B| = |{16, 17}| = 2 students.

c) To find the number of students only in the science class, we need to find the elements that are in set B but not in set A.

Number of students only in science class = |B - A| = |{21, 22}| = 2 students.

So, the answers are:

a) 3 students are in both classes.

b) 2 students are only in the math class.

c) 2 students are only in the science class.

This problem illustrates how sets can be used to analyze and solve real-world scenarios involving data collections, such as student ages in different classes.



Example.7) Suppose you have two sets, Set A and Set B, and you need to find their union and intersection. Set A contains the even numbers less than 10, and Set B contains prime numbers less than 10. Calculate A ∪ B (the union of A and B) and A ∩ B (the intersection of A and B).

Ans.)

First, let's define the sets:

Set A: {2, 4, 6, 8} (even numbers less than 10),

Set B: {2, 3, 5, 7} (prime numbers less than 10)

Now, we can calculate the union (A ∪ B) and the intersection (A ∩ B):-

Union (A ∪ B):- The union of two sets includes all distinct elements from both sets. In this case, it will contain all the even numbers less than 10 and all the prime numbers less than 10, without duplicates.

  A ∪ B = {2, 4, 6, 8, 3, 5, 7}

So, the union of Set A and Set B is {2, 4, 6, 8, 3, 5, 7}.

Intersection (A ∩ B):- The intersection of two sets contains only the elements that are common to both sets. In this case, it will include the numbers that are both even and prime, which are 2.

  A ∩ B = {2}

So, the intersection of Set A and Set B is {2}.

This is a basic example of solving problems involving sets, showing how to perform union and intersection operations on sets. Depending on the problem, you may encounter more complex operations, such as set differences, complements, or working with larger sets and set properties.