List the elements of the following sets: (i) ${x : x^{2} - 2 x - 3 = 0}$

(ii) ${x : x = 2 y + 5; y \in N a n d 2 \leq y < 6}$

(iii) ${x : x is a factor of 24}$

(iv) ${x : x \in Z a n d x^{2} \leq 4}$

(v) ${x : 3 x - 2 \leq 10, x \in N}$

(vi) ${x : 4 - 2 x > - 6, x \in Z}$

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Solution

(i) ${x : x^{2} 2 x - 3 = 0} x^{2} - 2 x - 3 = 0 \Rightarrow x^{2} - 3 x + x - 3 = 0 \Rightarrow x (x - 3) + 1 (x - 3) = 0 \Rightarrow (x + 1) (x - 3) = 0 ∴ E i t h e r x - 3 = 0 O r x + 1 = 0 x = 3 x = - 1 ∴ E l e m e n t s o f t h e s e t {x : x^{2} - 2 x - 3 = 0} a r e 3 a n d - 1 A n s .$

(ii) $x = 2 y + 5 W h e n y = 2, x = 2 \times 2 + 5 = 4 + 5 = 9 W h e n y = 3, x = 2 \times 3 + 5 = 6 + 5 = 11 W h e n y = 4, x = 2 \times 4 + 5 = 8 + 5 = 13 W h e n y = 5, x = 2 \times 5 + 5 = 10 + 5 = 15 ∴ E l e m e n t s o f t h e G i v e n s e t {x : x = 2 y + 5; y \in N a n d 2 \leq y < 6} a r e 9, 11, 13, 15 A n s .$

(iii) ${x : x is a factor of 24} 24 = 1 \times 24 24 = 2 \times 12 24 = 3 \times 8 24 = 4 \times 6 ∴ E l e m e n t s o f t h e g i v e n s e t x : x i s a f a c t o r o f 24 a r e 1, 2, 3, 4, 6, 8, 12, 24 A n s .$

(iv) ${x : x \in Z a n d x^{2} \leq 4} W h e n x^{2} = 4 x = \pm \sqrt{4} = \pm 2 W h e n x^{2} = 1 x = \pm \sqrt{1} = \pm 1 W h e n x^{2} = 0 x = \sqrt{0} = 0 ∴ Elements of the given set {x : x \in Z a n d x^{2} \leq 4} a r e + 2, - 2, + 1 - 1, 0 o r a r e - 2, - 1, 0, 1, 2 A n s$

(v) ${x : 3 x - 2 \leq 10, x \in N} 3 x - 2 \leq 10 \Rightarrow 3 x \leq 10 + 2 \Rightarrow 3 x \leq 12 \Rightarrow x \leq \frac{12}{3} \leq x \leq 4 ∴ Elements of the given set x : 3 x - 2 \leq 10, x \in N a r e 1, 2, 3 a n d 4 A n s .$

(vi) ${x : 4 - 2 x > - 6, x \in Z} 4 - 2 x > - 6 - 4 + 4 - 2 x > - 6 - 4 (Subtracting 4 from both sides) - 2 x > - 10 - 2 x + 2 x + 10 > - 10 + 2 x + 10 [Adding 2x+10 to both sides] + 10 > 2 x \frac{10}{2} > x 5 > x ∴ Elements of the given set {x : 4 - 2 x > - 6, x \in Z} a r e 4, 3, 2, 1, 0, - 1.................. A n s .$

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