Which of the following sets are equal? i. A = {1, 2, 3} ii. B = {x �

1.	Which of the following sets are equal? i. A = {1, 2, 3} ii. B = {x ∈ R:x2 – 2x +1 = 0} iii. C = (1, 2, 2, 3} iv. D = {x ∈R : x3 – 6x2+11x – 6 = 0}.
Answer» NOTE: A set is said to be equal with another set if all elements of both the sets are equal and same. A = {1, 2, 3} . B = {x ∈ R:x²– 2x + 1 = 0} x²– 2x + 1 = 0 (x–1)² = 0 ∴ x = 1. B = {1} C = {1, 2, 2, 3} In sets we do not repeat elements hence C can be written as {1, 2, 3} D = {x ∈R : x³ – 6x²+ 11x – 6 = 0}. For x = 1 = (1)³– 6(1)²+11(1) – 6 = 1 – 6 +11 – 6 = 0 For x =2 = (2)³– 6(2)²+ 11(2) – 6 = 8 – 24 + 22 – 6 = 0 For x =3 = (3)³– 6(3)²+ 11(3) – 6 = 27 – 54 + 33 – 6 = 0 As cubic equation has three roots at max so the roots are 1, 2, 3 ∴ D = {1, 2, 3} Hence Set A, C and D are equal.

Which of the following sets are equal? i. A = {1, 2, 3} ii. B = {x ∈ R:x2 – 2x +1 = 0} iii. C = (1, 2, 2, 3} iv. D = {x ∈R : x3 – 6x2+11x – 6 = 0}.

Answer»

NOTE: A set is said to be equal with another set if all elements of both the sets are equal and same.

A = {1, 2, 3} .

B = {x ∈ R:x²– 2x + 1 = 0}

x²– 2x + 1 = 0

(x–1)² = 0

∴ x = 1.

B = {1}

C = {1, 2, 2, 3}

In sets we do not repeat elements hence C can be written as {1, 2, 3}

D = {x ∈R : x³ – 6x²+ 11x – 6 = 0}.

For x = 1

= (1)³– 6(1)²+11(1) – 6

= 1 – 6 +11 – 6

= 0

For x =2

= (2)³– 6(2)²+ 11(2) – 6

= 8 – 24 + 22 – 6

= 0

For x =3

= (3)³– 6(3)²+ 11(3) – 6

= 27 – 54 + 33 – 6

= 0

As cubic equation has three roots at max so the roots are 1, 2, 3

∴ D = {1, 2, 3}

Hence Set A, C and D are equal.