Question 1: Which of the following are examples of empty set?

(i) Set of all even natural numbers divisible by $5$.

(ii) Set of all even prime numbers

(iii) $\{x : x^2 -2 = 0$ and $x$ is rational $\}$

(vi) $\{x : x$ is a natural number , $x < 8$ and simultaneously $x > 12\}$

(v) $\{x: x$ is a point common to any two parallel lines $\}$

(i) Set of all even natural numbers divisible by $5$ would mean a set $\{ x : x = 2 (5n), n \in N, n >0 \} \text{ or } \{ 10, 20, 30, 40, \cdots \}$. This has elements and therefore not an empty set.

(ii) Set of all even prime numbers would me a set $\{ 2 \}$. This has elements and therefore not an empty set.

(iii) $\{x : x^2 -2 = 0$ and $x$ is rational $\}$. This would mean that $x = \pm \sqrt{2}$. $\sqrt{2}$ is irrational number. Hence this is an empty set.

(vi) $\{x : x$ is a natural number , $x < 8$ and simultaneously $x > 12\}$. There are no number which are less than $8$ and also greater than $12$ simultaneously. Hence this is an empty set.

(v) $\{x: x$ is a point common to any two parallel lines $\}$. This is an empty set because two parallel lines never intersect.

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Question 2: Which of the following sets are finite and which are infinite?

(i) Set of concentric circles in a plane.

(ii) Set of letters of the English Alphabets.

(iii) $\{x \in N : x > 5\}$

(iv) $\{x \in N : x <200\}$

(v) $\{x \in Z : x < 5\}$

(vi) $\{x \in R : 0 < x < 1\}$

(i) Set of concentric circles in a plane: There can be infinite concentric circles. Therefore this is an INFINITE set.

(ii) Set of letters of the English Alphabets: There are only 26 English alphabets

$\{ \text{A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z } \}$

Hence this is a FINITE set.

(iii) $\{x \in N : x > 5\}$. This would be a set $\{ 6, 7, 8, \cdots \}$. There will be infinite elements in the set. Therefore this is an INFINITE set.

(iv) $\{x \in N : x <200\}$. This set will contain $\{ 1, 2, 3, 4, \cdots , 198, 199 \}$. Hence this is an FINITE set.

(v) $\{x \in Z : x < 5\}$. This set would be $\{ \cdots, -4, -3, -2, -1, 0, 1,2, 3, 4 \}$. There will be infinite elements in the set. Therefore this is an INFINITE set.

(vi) $\{x \in R : 0 < x < 1\}$. This set would  include all decimals, rational numbers and irrational numbers between $0$ and $1$. There will be infinite elements in the set. Therefore this is an INFINITE set.

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Question 3: Which of the following sets are equal?

(i) $A = \{1, 2, 3\}$

(ii) $B = \{ x \in R : x^2 -2x + 1 = 0\}$

(iii) $C = \{1, 2, 2, 3\}$

(vi) $D = \{x \in R : x^3 - 6x^2 + 11x -6=0\}$

Note: EQUAL SETS – Two sets $A$ and $B$ are said to be equal if every element of $A$ is a member of $B$, and every element of $B$ is a member of $A$.

Set $B = \{ x \in R : x^2 -2x + 1 = 0 \}$ is $\{ 1 \}$

$x^2 -2x + 1 = 0 \Rightarrow (x-1)^2 = 0 \Rightarrow x = 1$

$C = \{1, 2, 2, 3\}$ this can also be written as $C = \{1, 2, 3\}$ as repeating element can be eliminated.

$D = \{x \in R : x^3 - 6x^2 + 11x -6=0\}$ is $\{1, 2, 3, \}$

Hence $A = C = D$

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Question 4: Are the following sets equal?

(i) $A = \{x : x$ is a letter in the word reap $\}$

(ii) $B = \{x : x$ is a letter in the word paper $\}$

(iii) $C = \{x : x$ is a letter in the word rope $\}$

$A = \{x : x \text{ is a letter in the word reap } \} \text{ would mean } \{ r, e, a, p \}$

$B = \{x : x \text{ is a letter in the word paper } \} \text{ would mean } \{ p, a, e, r \}$

$C = \{x : x \text{ is a letter in the word rope } \} \text{ would mean } \{ r, o, p, e \}$

Therefore none of them are equal.

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Question 5: From the sets given below, find the pair the equivalent sets: $A = \{1, 2, 3\} , B = \{t, p, q, r, s\} , C = \{\alpha , \beta , \gamma \} , D = \{a, e, i, o ,u\}$

EQUIVALENT SETS: Two finite sets $A$ and $B$ are equivalent if their cardinal numbers are same. i.e. $n(A) = n(B)$.

Therefore $A = \{1, 2, 3\} \Rightarrow n(A) = 3$

$B = \{t, p, q, r, s\} \Rightarrow n(B) = 5$

$C = \{\alpha , \beta , \gamma \} \Rightarrow n(C) = 3$

$D = \{a, e, i, o ,u\} \Rightarrow n(D) = 5$

$'A \ and \ C'$  and $'B \ and \ C'$ are equivalent sets.

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Question 6: Are the following pairs of sets equal? Give reasons

(i) $A = \{2, 3\}$, $B = \{x : x$ is a solution of $x^2 + 5x + 6 = 0\}$

$\text{(ii) } A = \{x : x \text{ is a letter of the word 'WOLF' } \} \text{ , } \\ \\ B = \{x : x \text{ is a letter of the word 'FOLLOW' } \}$

(i) $A = \{2, 3\}$

$B = \{x : x$ is a solution of $x^2 + 5x + 6 = 0\} \Rightarrow \{ -2, -3 \}$

Therefore NOT EQUAL.

(ii) $A = \{x : x$ is a letter of the word $'WOLF' \}$ $\Rightarrow \{ W, O, L, F \}$

$B = \{x : x$ is a letter of the word $'FOLLOW' \}$ $\Rightarrow \{ F, O, L, W \} \ or \ \{ W, O, L, F \}$

Hence $A$ and $B$ are EQUAL.

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Question 7: From the sets given below, select equal sets and equivalent sets

$A = \{0, a\} B = \{1, 2, 3, 4\}, C = \{4, 8, 12\}, D = \{3, 1, 2, 4\}, E = \{1, 0\}, F = \{8, 4, 12\}, G = \{1, 5, 7, 11\} , H = \{a, b\}$

$A = \{0, a\} \Rightarrow n(A) = 2$          $B = \{1, 2, 3, 4\} \Rightarrow n(B) = 4$

$C = \{4, 8, 12\} \Rightarrow n(C) = 3$          $D = \{3, 1, 2, 4\} \Rightarrow n(D) = 4$

$E = \{1, 0\} \Rightarrow n(E) = 2$          $F = \{8, 4, 12\} \Rightarrow n(F) = 3$

$G = \{1, 5, 7, 11\} \Rightarrow n(G) = 4$          $H = \{a, b\} \Rightarrow n(H) = 2$

Hence we can see that $B$ and $D$ are equal. Also $C$ and $F$ are equal sets.

$A, E$ and $H$ are equivalent. Also $C$ and $F$ are equivalent sets. Also $D$ and $G$ are equivalent.

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Question 8: Which of the following sets are equal?

$A = \{ x: x \in N, x < 3\}, \ \ \ \ \ B = \{1, 2\}, C = \{3, 1\},$ $D = \{x : x \in N, x \ is \ odd \ , x < 5\}, \ \ \ E = \{1, 2, 1,1\}, \ \ \ F = \{1, 1, 3\}$

$A = \{ x: x \in N, x < 3\} \ or \ A = \{ 1, 2 \} \Rightarrow n(A) = 2$

$B = \{1, 2\} \Rightarrow n(B) = 2$

$C = \{3, 1\} \Rightarrow n(C) = 2$

$D = \{x : x \in N, x \ is \ odd \ , x < 5\} \ or \ D = \{1, 3 \} \Rightarrow n(D) = 2$

$E = \{1, 2, 1,1\} \Rightarrow n(E) = 2$

$F = \{1, 1, 3\} \Rightarrow n(F) = 2$

Therefore $A = B = E$ and $C = D + F$

They are also all equivalent sets as they all have $2$ elements each.

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Question 9: Show that the set of letters needed to spell $"CATARACT"$ and the set of letters needed to spell $"TRACT"$ are equal.

$\text{Set of letters needed to spell } "CATARACT" = \{ C, A, T, R \}$
$\text{Set of letters needed to spell } "TRACT" = \{ T, R, A, C \}$