NCERT Class 11 Mathematics — Sets, Exercise 1.2. All 6 questions solved.

Three ideas run through this exercise, and each is a yes-or-no test rather than a calculation.

• The empty set (or null set) $\varphi$ has no elements at all. A set is empty when its defining condition can be satisfied by nothing.
• A set is finite if its elements can be counted and the counting stops; otherwise it is infinite.
• Two sets are equal, written $A = B$, when they have exactly the same elements. Order and repetition are irrelevant — only membership counts.

Key insight. “Empty” and “small” are different claims, and so are “finite” and “listable”. $\{2\}$ is not empty just because it holds only one element, and the set of animals on Earth is finite even though nobody could ever list it. In each case the question is what the definition forces, not what is practical.

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) $\{y : y \text{ is a point common to any two parallel lines}\}$

Solution. A set is null when its condition is impossible to satisfy — so for each part, try to produce one element and see what stops you.

(i) Null. A number divisible by $2$ is even, and an odd number is by definition not even. The two requirements contradict each other, so nothing qualifies.

(ii) Not null. This is the part designed to catch you. $2$ is even and prime, so the set is $\{2\}$ — a one-element set, which is a very different object from $\varphi$. (It is, in fact, the only even prime, since every other even number has $2$ as a proper divisor.)

(iii) Null. No number is simultaneously less than $5$ and greater than $7$; the two inequalities have no overlap.

(iv) Null. Parallel lines are defined as lines that never meet, so there is no common point.

Question 2

Which of the following sets are finite or infinite?

(i) The set of months of a year (ii) $\{1, 2, 3, \ldots\}$ (iii) $\{1, 2, 3, \ldots 99, 100\}$ (iv) The set of positive integers greater than $100$ (v) The set of prime numbers less than $99$

Solution. Ask in each case whether the counting terminates.

(i) Finite — there are exactly $12$.

(ii) Infinite — this is $\mathbf{N}$ itself; the dots run on with no last term.

(iii) Finite — the dots here sit between a first and a last element, so the set has exactly $100$ members. Contrast with (ii): the position of the dots is what separates the two.

(iv) Infinite — $101, 102, 103, \ldots$ never stops.

(v) Finite — the primes below $99$ are bounded above, so there are only finitely many (twenty-five of them, from $2$ to $97$).

Question 3

State whether each of the following sets is finite or infinite:

(i) The set of lines which are parallel to the $x$-axis (ii) The set of letters in the English alphabet (iii) The set of numbers which are multiples of $5$ (iv) The set of animals living on the earth (v) The set of circles passing through the origin $(0,0)$

Solution.

(i) Infinite. A line parallel to the $x$-axis has equation $y = c$, and $c$ may be any real number. One line for every real number is uncountably many lines.

(ii) Finite. Twenty-six letters.

(iii) Infinite. $5, 10, 15, \ldots$ continues without end.

(iv) Finite. This is the part that provokes an argument. However enormous the number, it is a definite whole number — the counting would in principle terminate. Being impossible to carry out is not the same as being infinite.

(v) Infinite. A circle through the origin is fixed by its centre $(a, b)$, with radius $\sqrt{a^2+b^2}$ following automatically. Every point of the plane except the origin itself supplies a different such circle.

Question 4

In the following, state whether $A = B$ or not:

(i) $A = \{a, b, c, d\}$, $B = \{d, c, b, a\}$ (ii) $A = \{4, 8, 12, 16\}$, $B = \{8, 4, 16, 18\}$ (iii) $A = \{2, 4, 6, 8, 10\}$, $B = \{x : x \text{ is a positive even integer and } x \le 10\}$ (iv) $A = \{x : x \text{ is a multiple of } 10\}$, $B = \{10, 15, 20, 25, 30, \ldots\}$

Solution. Equality of sets is checked element by element, in both directions — every member of $A$ must be in $B$, and every member of $B$ in $A$. A single element on one side and not the other settles it.

(i) $A = B$. The same four letters, written in a different order. Order carries no information in a set.

(ii) $A \ne B$. $12 \in A$ but $12 \notin B$; equivalently $18 \in B$ but $18 \notin A$. Three shared elements out of four is not equality.

(iii) $A = B$. Expanding $B$: the positive even integers at most $10$ are $2, 4, 6, 8, 10$ — exactly $A$. The two sets are written in different forms, which is precisely why the roster-versus-set-builder work of Exercise 1.1 matters here.

(iv) $A \ne B$. $15 \in B$, but $15$ is not a multiple of $10$. ($B$ is the set of multiples of $5$ from $10$ onwards, which is a strictly larger collection.)

Question 5

Are the following pairs of sets equal? Give reasons.

(i) $A = \{2, 3\}$, $B = \{x : x \text{ is a solution of } x^2 + 5x + 6 = 0\}$ (ii) $A = \{x : x \text{ is a letter in the word FOLLOW}\}$, $B = \{y : y \text{ is a letter in the word WOLF}\}$

Solution.

(i) Solve the quadratic rather than guessing from the coefficients:

$$x^2 + 5x + 6 = (x+2)(x+3) = 0 \quad\Longrightarrow\quad x = -2,\ -3$$

So $B = \{-2, -3\}$, while $A = \{2, 3\}$. The signs are opposite throughout, so

$$A \ne B$$

The trap is real: the numbers $2$ and $3$ do appear in the factorisation, but as $-2$ and $-3$ in the roots. A positive-coefficient quadratic of this shape has negative roots.

(ii) FOLLOW has six letters, F, O, L, L, O, W, but only four distinct ones:

$$A = \{\mathrm{F, O, L, W}\}$$

WOLF has four letters, all distinct:

$$B = \{\mathrm{W, O, L, F}\}$$

Same four letters, so

$$A = B$$

Question 6

From the sets given below, select equal sets:

$$A = \{2, 4, 8, 12\},\quad B = \{1, 2, 3, 4\},\quad C = \{4, 8, 12, 14\},\quad D = \{3, 1, 4, 2\}$$ $$E = \{-1, 1\},\quad F = \{0, a\},\quad G = \{1, -1\},\quad H = \{0, 1\}$$

Solution. Group by size first — a two-element set can only equal another two-element set — then compare within each group.

Four-element sets: $A$, $B$, $C$, $D$.

• $B$ and $D$ hold the same four numbers $\{1, 2, 3, 4\}$, so $B = D$.
• $A$ and $C$ share $4, 8, 12$, but $A$ has $2$ where $C$ has $14$, so $A \ne C$.
• $A$ and $C$ cannot equal $B$ or $D$, since $8 \notin B$.

Two-element sets: $E$, $F$, $G$, $H$.

• $E$ and $G$ both hold $1$ and $-1$, so $E = G$.
• $F = \{0, a\}$ and $H = \{0, 1\}$ agree only if $a = 1$, and the question gives no such information. With $a$ an unspecified symbol, $F \ne H$.
• $E$ and $G$ contain $-1$, which is in neither $F$ nor $H$.

Common mistakes

• Question 1(ii), calling $\{2\}$ the null set. “Even prime” sounds self-contradictory, but $2$ satisfies both conditions. A set with one element is not empty — and $\{\varphi\}$ is not empty either, since it contains one thing.
• Question 2, reading dots without looking at what follows them. $\{1, 2, 3, \ldots\}$ is infinite and $\{1, 2, 3, \ldots, 100\}$ is finite. The presence of a final term after the dots is the entire difference.
• Question 3(iv), calling the animals on Earth infinite. Uncountable in practice is not infinite in mathematics. If a definite whole number answers “how many”, the set is finite.
• Question 5(i), reading the roots off the constant term. $x^2+5x+6$ factorises as $(x+2)(x+3)$, giving $x = -2, -3$. Writing $\{2, 3\}$ and declaring the sets equal is the single most common error in this exercise.
• Question 6, declaring $F = H$. They match only if $a = 1$, which is nowhere stated. An unknown symbol cannot be assumed to take a convenient value.
• Confusing equal sets with equivalent sets. $\{1, 2, 3\}$ and $\{a, b, c\}$ have the same number of elements but are not equal. Equality is about which elements, not how many.

Practise next

• Exercise 1.3 — subsets, the power set and intervals. The equality test used in questions 4 to 6 reappears there as the two-way containment $A \subset B$ and $B \subset A \Leftrightarrow A = B$.
• Exercise 1.1 — worth revising if the roster and set-builder conversions in question 4(iii) felt slow.