Comprehensive Course on Sets and Functions
1. Numbers and Basic Operations
The set of counting numbers starting from 0.
ℕ = {0, 1, 2, 3, 4, ...}
All positive, negative whole numbers, and zero.
ℤ = {..., -3, -2, -1, 0, 1, 2, 3, ...}
1.1 Arithmetic Operations
- Addition (+): Combining two or more numbers into a single number.
- Subtraction (-): Finding the difference between two numbers.
- Multiplication (×): Repeated addition of the same number.
- Division (÷): Repeated subtraction or splitting into equal parts.
- Modulo (mod): The remainder when one number is divided by another.
1. 5 + 2 = 7
2. 9 - 4 = 5
3. 3 × 4 = 12 (adding 3 four times)
4. 18 ÷ 3 = 6 (dividing 18 into 3 equal parts)
5. 10 mod 3 = 1 (when 10 is divided by 3, the remainder is 1)
Try These Problems:
- Calculate 15 mod 4.
- What is the result of 7 × 6?
- Simplify (8 + 2) × 3 - 6.
Solutions:
- 15 mod 4 = 3 (since 15 = 3 × 4 + 3)
- 7 × 6 = 42
- (8 + 2) × 3 - 6 = 10 × 3 - 6 = 30 - 6 = 24
1.2 Factors and Multiples
- a is a factor of b if b mod a = 0.
- b is a multiple of a.
1. 2 is a factor of 6 (since 6 mod 2 = 0)
2. 5 is a factor of 10 (since 10 mod 5 = 0)
3. 12 is a multiple of 3 (since 12 = 3 × 4)
1.3 Rational and Real Numbers
Numbers of the form $\frac{p}{q}$, where p, q ∈ ℤ and q ≠ 0.
1. $\frac{2}{5}$ is rational
2. $\frac{10}{20} = \frac{1}{2}$ is rational (after reduction)
3. 0.75 = $\frac{3}{4}$ is rational
Numbers that cannot be written as $\frac{p}{q}$ where p, q are integers.
Examples: $\sqrt{2}$, π, e
The set of all rational and irrational numbers.
1.4 Greatest Common Divisor (GCD)
The GCD of two non-zero integers p and q is the largest positive integer that divides both p and q without a remainder.
1. gcd(9, 12) = 3
2. gcd(15, 45) = 15
3. gcd(0, q) = q (for any non-zero integer q)
4. gcd(1, q) = 1 (for any integer q)
To find gcd(a, b):
1. If b = 0, then gcd(a, b) = |a|
2. Otherwise, gcd(a, b) = gcd(b, a mod b)
Try These Problems:
- Find gcd(18, 24)
- Find gcd(35, 42)
- Find gcd(17, 23)
Solutions:
- gcd(18, 24) = 6
- gcd(35, 42) = 7
- gcd(17, 23) = 1 (17 and 23 are coprime)
2. Sets
2.1 Definition and Notation
A set is a collection of well-defined objects, called elements or members of the set.
- Roster Form: Listing all elements within curly braces.
Example: A = {1, 2, 3, 4, 5} - Set Builder Form: Describing elements that satisfy certain conditions.
Example: B = {x | x is an even natural number less than 10}
1. The set of vowels in English: {a, e, i, o, u}
2. The set of even natural numbers less than 10: {0, 2, 4, 6, 8}
3. The set of perfect squares less than 30: {0, 1, 4, 9, 16, 25}
2.2 Cardinality
Related Chapters 📂
01 A Quick Introduction to Variables
In Python, a variable is essentially a name or a label that refers to a value or stored data. You can think of a variable as a basket in real life used to keep track of information in your program. Variables temporarily store data in the computer’s memory. Variables are created when they are first assigned a value. An assignment statement creates a new variable and gives it a value. For example, price = 10 creates a variable named price and assigns it the integer value 10. Similarly, x = 6 creates a variable x with the value 6.
🧠02 Activity Questions 1.1
1. Below is a list of numbers: 22, -17, 47, -2000, 0, 1, 43, 1729, 6174, -63, 100, 32, -9. How many natural numbers are there in the given list? a) 6 b) 7 c) 8 d) 9 Solution Based on the sources, the set of natural numbers is denoted by N. This set includes 0, 1, 2, 3, 4, 5, and so on. The sources explicitly state that whenever they are talking about natural numbers, it always includes a 0, even though some books may not. Natural numbers are primarily used for counting.
03 Activity Questions 1.1
“The cafe was like a battleship stripped for action.” The figure of speech used here is ___. Drawing on the information from the sources discussing figures of speech:
🧠04 Activity Questions 1.10
Answer the questions 1-3, based on following information: Let A = {x|x ∈ ℕ, x < 10 and x is odd} B = {y|y ∈ ℕ, y is a perfect square and 15 < y < 40} Q1. Which of the following is a subset of B × A? ○ {(36, 3), (25, 5), (36, 6)} ○ {(1, 25), (6, 36), (7, 25), (3, 36)} ○ {(16, 5), (25, 9), (36, 3), (16, 1)}
🧠05 Activity Questions 1.11
Q1. If Dom(f) = {x ∈ ℝ, f(x) ∈ ℝ} defined by f(x) = (x + 12)/(4x - 8), then the domain of the function f is ______ ○ ℝ ○ ℝ \ {1/4} ○ ℝ \ {-12} ○ ℝ \ {2} Solution Q2. The product of the minimum value of the function f(x) = 9|x| - 8 and the maximum value of the function g(x) = 11 - |x + 8| is ______
06 Activity Questions 1.2
Q1. Which among the following use alliteration in its name? a) Kolkata Knight Riders b) Peter Parker c) Big Billion Days d) All of the above Based on the information in the sources, alliteration is a literary device where the first sounds of two adjacent words or phrases are similar or the same. This repetition of the initial sound is used to make language more impactful and rhythmic. Examples from the sources include:
🧠07 Activity Questions 1.2
1. Which of the following option(s) is(are) true? Solution Based on the sources and our conversation history, we can determine which of the given inequalities between fractions are true by finding a common denominator and comparing the numerators. Rational numbers, which include fractions, can be written in the form p/q. To compare two fractions which have different denominators, there is no way to directly compare them. The only way is to convert them into equivalent fractions such that they have the same denominator. A number that is a multiple of both denominators can be used as the common denominator. Once the denominators are the same, you can add the numerators, or in this case, compare them.
🧠08 Activity Questions 1.3
1. Which of the following statement(s) is(are) false? a) The sum of two natural numbers is always a natural number b) The difference between two integers is always an integer c) The product of two rational numbers is always a real number d) The product of two irrational numbers is always an irrational number Solution The statement that is false is: