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Write the following sets in the roaster form:
(i) $D=\left\{t \mid t^{3}=t, t \in R\right\}$
(ii) $E=\left\{w \mid \frac{w-2}{w+3}=3, w \in \mathbf{R}\right\}$
(iii) $\mathrm{F}=\left\{\mathrm{x} \mid \mathrm{x}^{4}-5 \mathrm{x}^{2}+6=0, \mathrm{x} \in \mathbf{R}\right\}$
(i) According to the question,
D = {t | t3 = t, t ∈ R}
Roster form,
t3 = t
⇒ t3 – t = 0
⇒ t(t2 – 1) = 0
⇒ t(t – 1)(t + 1) = 0
⇒ t = 0, -1 or 1
Hence, D = {-1, 0, 1}
(ii) According to the question,
$E=\left\{w \mid \frac{w-2}{w+3}=3, w \in R\right\}$
Roster form,
((W – 2)/(W + 3))=3
⇒ w – 2 = 3(w + 3)
⇒ w – 2 = 3w + 9
⇒ 3w – w = – 9 – 2
⇒ 2w = –11
⇒ w = –11/2
Hence, E = {– 11/2}
(iii) According to the question,
F = {x | x4 – 5x2 + 6 = 0, x ∈ R}
Roster form,
x4 – 5x2 + 6 = 0
⇒ x4 – 3x2 – 2x2 + 6 = 0
⇒ x2(x2 – 3) – 2(x2 – 3) = 0
⇒ (x2 – 3) (x2 – 2) = 0
⇒ x2 = 3 or 2
⇒ x = ±√3 or ±√2
⇒ x = √3, –√3, √2 or –√2
Hence, F = {–√3, –√2, √2, √3}