CBSE Class 11 Maths: Sets Exercise 1.1 Explained

What is a Set? The Concept of a Well-Defined Collection

At its heart, a set is simply a collection of distinct objects. But not just any collection qualifies as a set in mathematics. The crucial condition is that the collection must be well-defined. What does "well-defined" mean? It means that for any given object, there must be no ambiguity as to whether that object belongs to the collection or not. There should be a clear, objective criterion for inclusion or exclusion.

For example, "the collection of all even numbers" is a well-defined set because we can definitively say if any number (like 2, 4, 100, or 7) is an even number or not. However, "the collection of the 5 best cricketers in India" is NOT a set, because the term "best" is subjective. What one person considers "best," another might not. There's no objective criterion. The individual objects in a set are called its elements or members. Sets are typically denoted by capital letters (A, B, C, etc.), and their elements by lowercase letters (a, b, c, etc.). Understanding this fundamental concept of a well-defined collection is the first and most important step in mastering sets.

Key Definitions for Sets

Set

A well-defined collection of distinct objects. The objects themselves are called elements or members of the set.

Well-defined

A collection is well-defined if it is absolutely clear and unambiguous whether a particular object belongs to the collection or not. Subjective criteria like 'good', 'best', 'tall' prevent a collection from being a set.

Elements / Members

The individual objects that constitute a set. For example, in the set of vowels {a, e, i, o, u}, 'a' is an element.

Empty Set (Null Set)

A set that contains no elements at all. It is denoted by the symbol $\emptyset$ or simply by {}. For example, the set of natural numbers less than 1 is an empty set.

Methods of Representing Sets: Roster Form and Set-Builder Form

Once we understand what a set is, the next step is to learn how to write them down clearly. There are primarily two ways to represent a set:

1. Roster or Tabular Form: In this method, all the elements of the set are listed, separated by commas, and enclosed within curly braces {}. The order in which the elements are listed does not matter, and elements are not repeated. For example, the set of all vowels in the English alphabet can be written as $V = \{a, e, i, o, u\}$. It could also be written as $V = \{e, a, u, i, o\}$, which represents the same set. Similarly, the set of letters forming the word 'SCHOOL' would be $S = \{S, C, H, O, L\}$ (note: 'O' is not repeated).

1. Set-Builder Form: This method describes the elements of a set by stating a common property that all elements of the set satisfy, and no other element outside the set satisfies. The general form is {x : P(x)}, where x represents an arbitrary element of the set and P(x) is the property or condition that x must satisfy. The colon : or a vertical bar | is read as "such that". For example, the set of all even natural numbers can be written as $E = \{x : x \text{ is an even natural number}\}$. The set $V = \{a, e, i, o, u\}$ could be written in set-builder form as $V = \{x : x \text{ is a vowel in the English alphabet}\}$. This form is particularly useful for describing sets with a large or infinite number of elements, where listing them all would be impractical or impossible.

Step-by-Step Examples: Mastering Set Representation

• Example 1: Identifying Sets Question: Which of the following collections are sets? Justify your answer. (i) The collection of all honest people in your city. (ii) The collection of all prime numbers less than 15. Solution: Step 1: Analyze collection (i). The term "honest" is subjective. What one person considers honest, another might not. There is no clear, objective criterion to determine if a person belongs to this collection or not. Step 2: Conclusion for (i). Therefore, "The collection of all honest people in your city" is not a set. Step 3: Analyze collection (ii). The property "prime number less than 15" is objective and well-defined. We can definitively determine if any number satisfies this property (e.g., 2, 3, 5, 7, 11, 13 are prime and less than 15; 4 or 17 are not). Step 4: Conclusion for (ii). Therefore, "The collection of all prime numbers less than 15" is a set, and it can be written in Roster form as {2, 3, 5, 7, 11, 13}.
• Example 2: Roster Form to Set-Builder Form Question: Write the set $A = \{1, 4, 9, 16, 25\}$ in set-builder form. Solution: Step 1: Observe the pattern in the elements. The elements are 1, 4, 9, 16, 25. These are the squares of natural numbers: $1^2=1$, $2^2=4$, $3^2=9$, $4^2=16$, $5^2=25$. Step 2: Identify the common property. All elements are squares of natural numbers, starting from 1 up to 5. Step 3: Formulate the property using 'x'. We can say 'x is the square of a natural number n, where n is less than or equal to 5'. Or, more concisely, 'x = n² where n is a natural number and $1 \le n \le 5
  . Step 4: Write in set-builder form. $A = \{x : x = n^2, \text{ where } n \in \mathbb{N} \text{ and } 1 \le n \le 5\}$.
• Example 3: Set-Builder Form to Roster Form Question: Write the set $B = \{x : x \text{ is an integer and } -2 < x \le 3\}$ in Roster form. Solution: Step 1: Understand the condition for 'x'. 'x' must be an integer, which includes positive numbers, negative numbers, and zero. Step 2: Apply the inequality condition. The condition is $-2 < x \le 3$. This means x must be greater than -2 but less than or equal to 3. Step 3: List all integers satisfying the condition. Integers greater than -2 are -1, 0, 1, 2, ... Integers less than or equal to 3 are ..., 1, 2, 3. Combining these, the integers that satisfy both conditions are -1, 0, 1, 2, 3. Step 4: Write the set in Roster form. $B = \{-1, 0, 1, 2, 3\}$.

Exam Tips for Sets Exercise 1.1

To ace your exams on Sets Exercise 1.1, keep these crucial points in mind:

• Focus on "Well-Defined": This is the most frequently tested concept for identifying whether a collection is a set. Always ask yourself: "Can everyone agree on whether an object belongs to this collection or not?" If the answer is no (due to subjectivity), it's not a set.
• Roster Form Rules: Remember that in Roster form:
• The order of elements does not matter (e.g., \{1, 2, 3\} is the same as \{3, 1, 2\}).
• Repetition of elements is ignored (e.g., \{1, 1, 2, 3\} is simply \{1, 2, 3\}). Only list distinct elements.
• Set-Builder Form Precision: When converting to or from Set-Builder form, be precise with your conditions. Use correct mathematical notation for number systems ($\mathbb{N}$ for natural numbers, $\mathbb{Z}$ for integers, $\mathbb{Q}$ for rational numbers, $\mathbb{R}$ for real numbers) and inequalities.
• Empty Set: Don't confuse \{\emptyset\} with $\emptyset$. The former is a set containing one element (the empty set itself), while the latter is an empty set. The empty set contains no elements. An empty set is always a well-defined set.

Practice Questions with Solutions

• Q: Which of the following collections are sets? Justify your answer: (i) The collection of all difficult questions in this chapter. (ii) The collection of all natural numbers less than 10. A: Step 1: Analyze collection (i). The term "difficult" is subjective; what is difficult for one student might be easy for another. There is no objective criterion for defining a difficult question. Step 2: Conclusion for (i). Therefore, (i) is NOT a set. Step 3: Analyze collection (ii). The property "natural numbers less than 10" is objective and unambiguous. We can list the elements clearly (1, 2, 3, 4, 5, 6, 7, 8, 9). Step 4: Conclusion for (ii). Therefore, (ii) IS a set. Final answer: (i) Not a set, (ii) Is a set.
• Q: Write the set $A = \{x : x \text{ is an integer, } x^2 \le 9\}$ in Roster form. A: Step 1: Understand the conditions for 'x'. 'x' must be an integer, and its square must be less than or equal to 9. Step 2: Find integers whose squares satisfy $x^2 \le 9$. If $x=0, x^2=0 \le 9$. (Valid) If $x=1, x^2=1 \le 9$. (Valid) If $x=-1, x^2=1 \le 9$. (Valid) If $x=2, x^2=4 \le 9$. (Valid) If $x=-2, x^2=4 \le 9$. (Valid) If $x=3, x^2=9 \le 9$. (Valid) If $x=-3, x^2=9 \le 9$. (Valid) If $x=4, x^2=16 \not\le 9$. (Not valid) Step 3: List the valid integers. The integers satisfying the condition are -3, -2, -1, 0, 1, 2, 3. Final answer: $A = \{-3, -2, -1, 0, 1, 2, 3\}$.
• Q: Write the set $B = \{0, 3, 6, 9, 12, 15\}$ in Set-Builder form. A: Step 1: Observe the pattern in the elements. The elements are multiples of 3, starting from 0. Step 2: Identify the range of multiples. $0 = 3 \times 0$, $3 = 3 \times 1$, $6 = 3 \times 2$, ..., $15 = 3 \times 5$. Step 3: Formulate the property. Each element 'x' is a multiple of 3, such that $x = 3n$, where 'n' is a whole number (non-negative integer) from 0 to 5. Final answer: $B = \{x : x = 3n, \text{ where } n \in \mathbb{W} \text{ and } 0 \le n \le 5\}$ (or $n \in \mathbb{Z}, 0 \le n \le 5$).
• Q: List all elements of the set $C = \{x : x \text{ is a consonant in the English alphabet before the letter 'f'}\}$. A: Step 1: Identify consonants in the English alphabet. Consonants are letters that are not vowels (a, e, i, o, u). Step 2: List letters in the English alphabet before 'f'. These are a, b, c, d, e. Step 3: From the list in Step 2, identify which ones are consonants. 'a' is a vowel, 'b' is a consonant, 'c' is a consonant, 'd' is a consonant, 'e' is a vowel. Step 4: Collect the identified consonants. Final answer: $C = \{b, c, d\}$.
• Q: Determine if the following statement is True or False: If $A = \{x, y, z\}$ and $B = \{z, x, y\}$, then $A = B$. A: Step 1: Recall the property of sets in Roster form. The order of elements in a set does not change the set itself. Step 2: Compare the elements of set A and set B. Both sets contain exactly the same distinct elements: x, y, and z. Step 3: Apply the property. Since the order of elements doesn't matter, and both sets have identical elements, they represent the same set. Final answer: True.

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