GSEB Class 11 Maths Solutions Chapter 1 Sets Exercise 1.2

Question 1. Which of the following are examples of the null set?
(i) Set of odd natural numbers divisible by 2.
(ii) Set of even prime numbers.
(iii) \( \{x : x \text{ is a natural number, } x < 5 \text{ and } x > 7\} \)
(iv) \( \{y : y \text{ is point common to any parallel lines}\} \)
Answer: The following options give examples of the null set:
(i) There is no odd natural number that can be divided evenly by 2. All odd numbers are not divisible by 2.
(iii) There is no natural number \( x \) that is both smaller than 5 and larger than 7 at the same time. These conditions cannot be met simultaneously.
(iv) Parallel lines never meet or cross at any point, so there is no common point between them.
Note: For option (ii), there is one even prime number, which is 2. Because of this, the set of even prime numbers is not an empty set; it contains the number 2.
In simple words: A null set means a set with no items inside it. We look at each option to see if it's possible to find any items that fit the description. If no items exist, it's a null set.

Exam Tip: Remember that a null set (or empty set) contains no elements. Always check if there's even one element that satisfies the given condition; if so, it's not a null set.

Question 2. Which of the following sets are finite or infinite?
1. The set of months of a year.
2. \( \{1, 2, 3, ...\} \)
3. \( \{1, 2, 3, ..., 99,100\} \)
4. The set of positive integers greater than 100.
5. The set of prime numbers less than 99.
Answer:
1. The set of months of a year: This is a finite set because there are exactly 12 members in this set (January, February, etc.), representing the months of a year.
2. \( \{1, 2, 3, ...\} \): This is an infinite set as it contains an endless number of natural numbers.
3. \( \{1, 2, 3, ..., 99,100\} \): This is a finite set because it contains precisely the first 100 natural numbers.
4. The set of positive integers greater than 100: This is an infinite set since there are an endless number of positive integers, such as 101, 102, 103, and so on, that are larger than 100.
5. The set of prime numbers less than 99: This is a finite set, because the set contains specific prime numbers like \( \{2, 3, 5, 7, ..., 97\} \). The count of these numbers is definite.
In simple words: A finite set has a countable number of items, while an infinite set has an unlimited number of items. Count how many items are in each set to decide.

Exam Tip: Look for ellipses (...) at the end of a set definition to quickly identify infinite sets, unless the ellipses are followed by an end number, which means it's a finite, bounded sequence.

Question 3. State whether each of the following sets is finite or infinite:
1. The set of lines which are parallel to the x-axis.
2. The set of letters in English alphabet.
3. The set of numbers which are multiples of 5.
4. The set of animals living on earth.
5. The set of circles through origin [0,0].
Answer:
1. Infinite: An unlimited number of lines can be drawn that are parallel to the x-axis. We can draw them at any y-coordinate, and there are infinite y-coordinates.
2. Finite: This set consists of 26 letters, which is a fixed and countable number.
3. Infinite: The multiples of 5, such as \( \{5, 10, 15, ...\} \), continue forever without end. Therefore, it is an infinite set.
4. Finite: While large, there is a definite, countable number of animals living on Earth. This number changes, but it is always a specific, finite count at any given moment.
5. Infinite: An unlimited number of circles can be drawn that pass through the origin point \( [0,0] \). You can have circles of different sizes and positions all passing through that single point.
In simple words: Decide if you can count all the items in the set. If you can, it's finite. If the items go on forever, it's infinite.

Exam Tip: Be careful with sets that represent real-world objects, like "animals on Earth." Even if the number is huge, if it's not endless, it's considered finite.

Question 4. In the following, state whether A = B or not:
1. A = \( \{a, b, c, d\} \) B = \( \{d, c, b, a\} \)
2. A = \( \{4, 8, 12, 16\} \) B = \( \{8, 4, 16, 18\} \)
3. A = \( \{2, 4, 6, 8, 10\} \) B = \( \{x : x \text{ is positive even integer and } x \le 10\} \)
4. A = \( \{x: x \text{ is multiple of } 10\} \) B = \( \{10, 15, 20, 25, 30, ...\} \)
Answer:
1. A = B: Sets A and B are equal because they contain the exact same elements. The order of elements within a set does not matter.
2. A \( \neq \) B: Sets A and B are not equal because set B contains the element 18, which is not present in set A. Also, set A contains 12, which is not in B.
3. A = B: Set A is \( \{2, 4, 6, 8, 10\} \). Set B can be written out as all positive even integers less than or equal to 10, which means B = \( \{2, 4, 6, 8, 10\} \). Since both sets have the same elements, A = B.
4. A \( \neq \) B: Set A consists of multiples of 10: \( \{10, 20, 30, 40, ...\} \). Set B includes numbers like 15 and 25, which are not multiples of 10. Since B contains elements not found in A, they are not equal.
In simple words: Two sets are equal if they have exactly the same items inside them, no matter the order. If even one item is different, or if one set has an item the other doesn't, they are not equal.

Exam Tip: When comparing sets, always list out the elements of both sets explicitly, especially when one is defined by a rule, to easily see if all elements match.

Question 5. Are the following pair of sets equal? Give reasons?
(i) A = \( \{2, 3\} \) B = \( \{x: x \text{ is a solution of } x^2 + 5x + 6 = 0\} \).
(ii) A = \( \{x : x \text{ is a letter in the word FOLLOW}\} \)
B = \( \{y : y \text{ is a letter in the word WOLF}\} \).
Answer:
(i) No, the sets are not equal. To find the elements of set B, we need to solve the quadratic equation \( x^2 + 5x + 6 = 0 \).
Factoring the equation gives us:
\( (x + 2)(x + 3) = 0 \)
This means the solutions for \( x \) are:
\( x = -2 \text{ or } x = -3 \)
So, set B = \( \{-2, -3\} \).
Clearly, set A = \( \{2, 3\} \) and set B = \( \{-2, -3\} \) are different because their elements are not the same.
(ii) Yes, the sets are equal. For set A, the unique letters in the word FOLLOW are F, O, L, W. So, A = \( \{F, O, L, W\} \). For set B, the unique letters in the word WOLF are W, O, L, F. So, B = \( \{W, O, L, F\} \). Since every element found in A is also present in B, and every element in B is also present in A, the sets are equal.
In simple words: For part (i), we solve the equation to find set B, then compare it to set A. For part (ii), we list the unique letters in each word to form the sets, then compare them. If all items are the same, the sets are equal.

Exam Tip: When letters from a word form a set, remember to only include unique letters once. The order of letters in the word, or in the resulting set, does not affect set equality.

Question 6. From the sets, given below, select examples of equal sets:
A = \( \{2, 4, 8, 12\} \)
B = \( \{1, 2, 3, 4\} \)
C = \( \{4, 8, 12, 14\} \)
D = \( \{3, 1, 4, 2\} \)
E = \( \{-1, 1\} \)
F = \( \{0, 0\} \)
G = \( \{1, -1\} \)
H = \( \{0, 1\} \)
Answer: The equal sets among the given options are B and D, and E and G.
Let's examine why:
Set B = \( \{1, 2, 3, 4\} \)
Set D = \( \{3, 1, 4, 2\} \)
Since the order of elements in a set does not matter, B and D contain the same elements, making them equal sets.
Set E = \( \{-1, 1\} \)
Set G = \( \{1, -1\} \)
Similarly, E and G also contain the same elements, just in a different order, so they are equal sets.
In simple words: To find equal sets, look for sets that have exactly the same numbers or items inside them. The order of the items doesn't matter, just that every item is identical.

Exam Tip: Always write out the elements of each set clearly. For sets with rules, interpret the rule to list its elements. For numerical sets, check if every number in one set is also in the other, and vice versa.