Set Theory and Number System

Set Theory and Number systems form the bedrock of modern mathematics, providing a systematic framework for understanding and manipulating mathematical concepts. Set theory deals with collections of objects, known as sets, and their relationships, while number systems classify and represent numbers in various ways. These foundational topics are essential for solving complex mathematical problems and are widely applied in diverse fields such as physics, computer science, and engineering.

1.0Definition of Set Theory

Set theory is a mathematical branch that studies collections of objects, called sets, and the relationships between these sets. In set theory, a set is defined as a well-defined collection of distinct objects, considered a single entity. The objects within a set are called elements or members. Set theory provides tools and techniques to analyze and manipulate sets, including operations such as union, intersection, complement, and power set construction. It forms the foundation of mathematics and is widely used in various areas of mathematics, computer science, and other disciplines for modeling and solving problems.

2.0Basic Concepts of Set Theory

Set theory is an essential and fundamental branch of mathematics that serves as the foundation for studying collections of objects, known as sets, and the relationships between these sets. Here are the basic concepts of set theory:

1. Sets: A set is a well-defined collection of distinct objects called elements. Sets are denoted by curly braces { }, and elements are listed inside these braces enclosed elements separated by commas. For instance, consider the set of integers that are larger than 5 and smaller than 10 can be represented as {6, 7, 8, 9}.
2. Elements: The objects within a set are called as elements of a set. Ex:- {1, 2, 3}, Set A three elements. 
3. Subset: A set A is said to be a subset of set B if all elements in A are also in B, denoted as A ⊆ B. For instance, {1, 2} is a subset of {1, 2, 3}.
4. Universal Set: The universal set, denoted as U, is the set that contains all possible elements under consideration in a particular context.
5. Empty Set: The empty set, denoted as ϕ or { }, is the set with no elements. It is a subset of every set.
6. Union: The union/combination of two sets, A and B, denoted as A ∪ B, is the set that contains all elements that are in A, in B, or in both A and B.
7. Intersection: The intersection/overlap of two sets, A and B, denoted as A ∩ B, is the set that contains all elements that are common to both A and B.
8. Complement: The complement of a set A, written as A′ or $\overline{\mathrm{A}}$, is the set of all elements in the universal set that are not in A.
9. Power Set: The power set of a set A, denoted as P(A), is the set of all subsets of A, including the empty set and A itself.

3.0Types of Sets

1. Null Set, or Empty Set, or Void Set

A set having no elements is called a Null Set or Empty Set or Void set. It is denoted by ϕ or { } .

2. Singleton Set

A set consisting of one or a single element is called a Singleton Set.

3. Finite Set

A set that has only a finite number of elements is called a Finite Set.

4. Infinite Set

A set which has an infinite number of elements is called an infinite Set.

5. Subset

If every element in set A is also found in set B, then A is termed as a subset of B, denoted as A ⊆ B.

6. Proper subset

If set A is a subset of set B, but A is not equal to B, then A is considered a proper subset of B, represented as A ⊂ B.

7. Universal Set

A set consisting of all possible elements that occur in the discussion is called a Universal set and is denoted by U.

8. Power set

Let A be any set. The set of all subsets of A is called the power set of A and is denoted by P(A).

9. Equal Set

If every element of set A is also an element of set B, and vice versa, they are equal sets denoted as A = B. 

For example, A = {3,4,5,6} and B = {6,5,4,3} are equal sets. However, A = {set of even numbers} and B = {set of natural numbers} are not equal as their elements differ.

10. Equivalent Set

If no. of elements in Set A is equal to the no. of elements in Set B, then Set A and Set B are called as equivalent sets.

Ex: A = {1, 2, 3, 4} and B = {w, x, y, z,}, n(A) = n(B) = 4. A and B are equivalent Sets; but A ≠ B.

11. Disjoint Sets

Two sets, X and Y, are called disjoint sets if they have no common elements, denoted as X ∩ Y = ϕ. This signifies that their intersection results in zero elements.

4.0Set Theory Venn Diagrams

Venn diagrams are powerful visual tools used in set theory to represent the relationships between sets and their elements.

Some Venn diagrams are given below-

5.0Properties of Sets 

1. Associative Property

• A ∪ (B ∪ C) = (A ∪ B) ∪ C
• A ∩ (B ∩ C) = (A ∩ B) ∩ C

2. Distributive Property

• A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
• A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

3. De Morgan’s Law

• Law of union : ( A ∪ B)′ = A′ ∩ B′ 
• Law of Intersection : ( A ∩ B)′ = A′ ∪ B′

4. Complement Law

• A ∪ A′ = A′ ∪ A = U
• A ∩ A′ = ϕ

6.0Set Theory Formula

Some of the Important Sets Formula are-

• n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
• n(A ∪ B) = n(A) + n(B) ⇔ A, B are disjoint sets.
• n(A – B) + n( A ∩ B ) = n(A)
• n(B – A) + n( A ∩ B ) = n(B)
• n( A – B) + n ( A ∩ B) + n( B – A) = n ( A ∪ B )
• n ( A ∪ B ∪ C ) = n(A) + n(B) + n(C) – n ( A ∩ B) – n ( B ∩ C) – n ( C ∩ A) +  n ( A ∩ B  ∩ C)

7.0Number System

Number systems are fundamental structures in mathematics that provide a way to represent and manipulate numbers. Here are the key number systems:

1. Natural Numbers (N)

Natural numbers are positive integers beginning from 1 and extending infinitely (1, 2, 3, 4, ...)

2. Whole Numbers (W)

Whole numbers are similar to natural numbers but include zero (0, 1, 2, 3, ...).

3. Integers (Z)  

Integers are whole numbers, including positive numbers, negative numbers, and zero. They are denoted by the symbol "Z" in mathematics. Examples of integers include -3, -2, -1, 0, 1, 2, 3, and so on.

4. Rational Numbers (Q)

Rational numbers are those that can be expressed as fractions of two integers where the denominator is not equal to zero (e.g., 1/2, -3/4, 5, -7, 0).

5. Real Numbers (R)

Real numbers encompass both irrational and rational numbers. They are numbers that can be located on the number line and include integers, fractions, decimals, and square roots of non-negative numbers. Examples of real numbers are -3, 0, 1.5, \sqrt{2}, and π.

6. Complex Numbers (C)

Complex numbers are mathematical entities of the form a + bi, where "a" and "b" are real numbers and "i" represents the imaginary unit (\sqrt{(-1)}). Complex numbers can be represented on a complex plane.

It may be noted that N ⊂ Z ⊂ Q ⊂ R ⊂ C.

7. Prime Number

Natural number having exactly two positive divisors. I.e., 1 and itself are called prime numbers. (e.g., 2, 3, 5, 7, 11, etc)

8. Composite Number

A composite no is a positive integer greater than 1 that has more than two factors, excluding 1 and itself. (e.g., 4, 6, 8, 9, etc)

9. Co-prime Number

Co-prime numbers are two numbers that have no common factors other than 1, i.e., their greatest common divisor (GCD) is 1. Example,  (1, 2), (1, 3), (7, 8) are co-prime numbers.

Each number system has its properties, operations, and applications in mathematics and other fields. Understanding these number systems is crucial for various mathematical concepts, calculations, and problem-solving techniques.

8.0Divisibility Test

9.0Set Theory Questions

Question 1. If A = {1, 2} then find its power set.

Solution: A = {1, 2} then P(A) = {Φ, {1}, {2}, {1, 2}}

Question 2. Show that A ∪ B = A ∩ B implies A = B

Solution: Let a ∈ A. Then a ∈ A ∪ B. Since A ∪ B = A ∩ B, a ∈ A ∩ B. So a ∈ B.

Therefore, A ⊂ B, Similarly, if b ∈ B, then b ∈ A ∪ B. Since

⇒ Α ∪ Β = A ∩ B, b ∈ A ∩ B. So, b ∈ A. Therefore, B ⊂ Α. Thus, A = B

Question 3. In a group of 1000 people, there are 750 who can speak Hindi and 400 who can speak Bengali. How many can speak Hindi only? How many can speak Bengali? How many can speak both Hindi and Bengali?

Ans. (250)

Solution: Let A and B be the sets of persons who can speak Hindi and Bengali respectively.

then n(A ∪ B) = 1000, n(A) = 750, n(B) = 400.

Number of persons whose can speak both Hindi and Bengali

⇒ n(A ∩ B) = n(A) + n(B) – n(A ∪ B)

⇒ 750 + 400 – 1000 = 150

Number of persons who can speak Hindi only. 

⇒ n(A – B) = n(A) – n(A ∩ B) = 750 – 150 = 600

Number of persons who can speak Bengali only.

⇒ n(B – A) = n(B) – n(A ∩ Β) = 400 – 150 = 250

Question 4. If A, B and C are three sets such that A ∩ B = A ∩ C and A ∪ B = A ∪ C, then:

(1) B = C (2) A ∩ B = φ (3) A = B (4) A = C

Ans. (1)

Solution: We have, A ∪ B = A ∪ C ⇒  (A ∪ B) ∩ C = (A ∪ C) ∩ C

⇒ (A ∩ C) ∪ (B ∩ C) = C [∵   (A ∪ C) ∩ C = C]

⇒ (A ∩ B) ∪ (B ∩ C) = C ...(i) [∵  A ∩ C = A ∩ B]

Again, A ∪ B = A ∪ C

⇒ (A ∪ B) ∩ B = (A ∪ C) ∩ B ⇒ B = (A ∩ B) ∪ (C ∩ B)

⇒ (A ∩ B) ∪ (C ∩ B) = B ⇒ (A ∩ B) ∪ (B ∩ C) = B ...(ii)

From (i) and (ii), we get B = C 

Question 5. In a class of 140 students numbered 1 to 140, all even numbered students opted mathematics course, those whose number is divisible by 3 opted Physics course and those whose number is divisible by 5 opted Chemistry course. Then the number of students who did not opt for any of the three courses is:         

(1) 102 (2) 42 (3) 1 (4) 38

Ans. (4)

Solution:

Let n(A) = number of students opted Mathematics = 70,

n(B) = number of students opted Physics = 46,

n(C) = number of students opted Chemistry = 28,

n(A ∩ B) = 23,

n(B ∩ C) = 9,

n(A ∩ C) = 14,

n(A ∩ B ∩ C) = 4,

Now n(A ∪ B ∪ C) 

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

= 70 + 46 + 28 – 23 – 9 – 14 + 4 = 102

So, number of students not opted for any course

= Total – n(A ∪ B ∪ C) 

= 140 – 102 = 38.

10.0Number System Questions

Question 1. Consider a number of N = 214534Q, then find number of values of Q so that N is divisible by 2

(A) 2 (B) 3 (C) 4 (D) 5

Ans. (D)

Solution:

For a number to be divisible by 2. The digit at the unit's place should be divisible by 2. In the number 'N' digit at unit place is Q. So, Q digit be such that it is divisible by 2.

⇒    Q = 0, 2, 4, 6, 8. 

Number of values of Q is 5.

Question 2. Consider a number N = 21453Q4. Number of values of Q so that number 'N' is divisible by 8, is 

(A) 4 (B) 3 (C) 2 (D) 6

Ans. (B)

Solution: N is divisible by 8 if the last three digits of the number all together are divisible by 8.

Here last three digit are 3Q4

3Q4 should be divisible by 8.

for Q = 0, the last three digits are 304 which is divisible by 8.

Similarly, only for Q = 4, 8 last three will be divisible by 8. 

⇒ Q = 0, 4, 8

The number of values Q is 3.