Sets Ex 1.2 Class 11 NCERT: Empty, Finite, Infinite, and Equal Sets

Key Definitions for Sets Ex 1.2

The Empty Set (Null Set)

A set which does not contain any element is called the empty set or the null set. It is denoted by the symbol ∅ or {}. For example, the set of whole numbers less than 0 is an empty set.

Finite Set

A set is called finite if it is either empty or consists of a definite, countable number of elements. For example, the set of vowels in the English alphabet, V = {a, e, i, o, u}, is a finite set as it has 5 elements.

Infinite Set

A set that is not finite is called an infinite set. The process of counting the elements in such a set would never end. For example, the set of all natural numbers, N = {1, 2, 3, ...}, is an infinite set.

Equal Sets

Two sets, A and B, are said to be equal if they have exactly the same elements. We write this as A = B. The order in which the elements are listed does not matter. For example, if A = {1, 2, 3} and B = {3, 1, 2}, then A = B.

Understanding the Difference: Finite vs. Infinite Sets

Distinguishing between finite and infinite sets is a fundamental skill. The core question to ask is: "Can I, in principle, finish counting the elements?" If the answer is yes, the set is finite. If the counting process would continue forever, it's infinite.

Consider the set of all people living in India. While this is a very large number, it is a specific, countable number. So, the set is finite. Now, consider the set of all points on a line segment connecting points (0,0) and (1,0). You can pick 0.1, 0.11, 0.111, and so on. There is no end to the number of points you can find. This is an infinite set.

When sets are given in set-builder form, analyze the condition. For example, A = {x : x ∈ N and x < 1000}. This set contains natural numbers from 1 to 999. It has a definite number of elements (999), so it's finite. In contrast, B = {x : x ∈ N and x > 1000} contains all natural numbers starting from 1001 and going on forever (1001, 1002, ...). This set is infinite.

Worked Examples from NCERT Concepts

• Question 1: Are the sets A = {-2, 2} and B = {x : x is a real number and x² - 4 = 0} equal? Step 1: Identify the elements of Set A. Set A is given in roster form. Its elements are clearly A = {-2, 2}. Step 2: Find the elements of Set B by solving the given condition. The condition for Set B is the equation x² - 4 = 0. To solve for x, we can add 4 to both sides: x² = 4. Taking the square root of both sides gives x = ±√4. So, x = 2 or x = -2. Therefore, the elements of Set B in roster form are B = {-2, 2}. Step 3: Compare the elements of Set A and Set B. Set A = {-2, 2} and Set B = {-2, 2}. Both sets contain exactly the same elements. Final Answer: Yes, the sets A and B are equal (A = B).
• Question 2: Which of the following is an example of a null set? (i) Set of odd natural numbers divisible by 2. (ii) Set of even prime numbers. Step 1: Analyze the condition for set (i). Odd natural numbers are {1, 3, 5, 7, ...}. A number is divisible by 2 if it is even. By definition, an odd number cannot be even. Therefore, there are no odd natural numbers that are divisible by 2. This set contains no elements. Step 2: Analyze the condition for set (ii). Prime numbers are {2, 3, 5, 7, 11, ...}. An even number is any integer divisible by 2. We need to find a number that is in both lists. The number 2 is a prime number, and it is also an even number. This set contains one element: {2}. Final Answer: The set of odd natural numbers divisible by 2 is a null (empty) set. The set of even prime numbers is not a null set; it is {2}.
• Question 3: Classify the following set as finite or infinite: The set of lines which are parallel to the y-axis. Step 1: Understand the geometric condition. The y-axis is a vertical line on the Cartesian plane. Lines parallel to the y-axis are also vertical lines. Step 2: Represent these lines with equations. Any vertical line can be represented by the equation x = k, where k is a constant real number. For example, x = 1, x = 2, x = -5.3, x = π are all lines parallel to the y-axis. Step 3: Determine if the number of such lines is countable or endless. Since 'k' can be any real number, there are infinitely many choices for k. We can have x = 1, x = 1.1, x = 1.11, x = 1.111, and so on. You can always find another real number between any two given real numbers. Because there is no end to the number of possible values for k, there is no end to the number of lines parallel to the y-axis. Final Answer: The set of lines which are parallel to the y-axis is an infinite set.

Exam Traps and Common Mistakes

Be careful with these common points of confusion in your exams:

• Empty Set vs. Zero Set: The empty set ∅ or {} has no elements. The set {0} is not empty; it contains one element, which is the number 0. Similarly, {∅} is also not empty; it's a set containing one element, that element being the empty set itself.
• Equal Sets vs. Equivalent Sets: Exercise 1.2 focuses on equal sets, which must have the exact same elements. Later, you'll learn about equivalent sets, which simply have the same number of elements. For example, A = {1, 2, 3} and B = {a, b, c} are equivalent (both have 3 elements) but they are not equal.
• Repetition and Order in Equal Sets: Remember that when comparing sets for equality, order does not matter and repetitions are ignored. The set {F, O, L, L, O, W} is written in roster form as {F, O, L, W}. This is exactly equal to the set {W, O, L, F}.

Practice Questions with Solutions

• Q: Which of the following sets is finite and which is infinite? (i) The set of months in a year. (ii) {1, 2, 3, ...} (iii) {x | x ∈ N and x is an odd number} A: Step 1: Analyze set (i). The number of months in a year is fixed at 12. This is a definite, countable number. Answer (i): Finite. Step 2: Analyze set (ii). The notation '...' indicates that the list of natural numbers continues without end. Answer (ii): Infinite. Step 3: Analyze set (iii). This represents the set of odd natural numbers {1, 3, 5, 7, ...}. This list also goes on forever. Answer (iii): Infinite.
• Q: Are the sets A = {a, b, c, d} and B = {d, c, b, a} equal? A: Step 1: Identify the elements of set A. The elements are a, b, c, and d. Step 2: Identify the elements of set B. The elements are d, c, b, and a. Step 3: Compare the elements. Both sets contain the exact same four elements. The property of sets states that the order of elements does not matter for equality. Final Answer: Yes, A = B.
• Q: Is the set A = {x : x is a natural number, x < 5 and x > 7} an empty set? A: Step 1: Analyze the conditions for the elements of set A. The element 'x' must be a natural number. Step 2: Apply the two conditions simultaneously. The element 'x' must be less than 5 AND greater than 7 at the same time. Step 3: Check if any number can satisfy both conditions. It is impossible for a number to be both smaller than 5 and larger than 7. Therefore, there are no elements that can belong to this set. Final Answer: Yes, set A is an empty set.
• Q: Are the sets A = {letters in the word 'LOYAL'} and B = {letters in the word 'ALLOY'} equal? A: Step 1: Write Set A in roster form. The distinct letters in 'LOYAL' are L, O, Y, A. So, A = {L, O, Y, A}. Step 2: Write Set B in roster form. The distinct letters in 'ALLOY' are A, L, O, Y. So, B = {A, L, O, Y}. Step 3: Compare the elements of the two sets. Both sets contain the same four letters: A, L, O, and Y. The order is different, but for sets, order does not matter. Final Answer: Yes, A = B because they contain the exact same elements.

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