Question

Let U = {x / x = 2", n E W, n<8) be the universal set.<br />A = {yly = 4", n E W, n<4}<br />B = {zız = 8", n E W,n<2}<br />then find (i) A' (ii) B' (iii) (A n B)'.​

Answer 1

Question:

Let  $U=\{x\mid x=2^n,\ n\in\mathbb{W},\ n<8\}$  be the universal set.

There exists two sets    such that,

$A=\{y\mid y=4^n,\ n\in\mathbb{W},\ n<4\}$

$B=\{z\mid z=8^n,\ n\in\mathbb{W},\ n<2\}$

Then find,

$(i)\ \ A'\\ \\ (ii)\ \ B'\\ \\ (iii)\ \ (A\cap B)'$

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[Note:  $\mathbb{W}$  is the set of whole numbers.]

$\mathbb{W}=\{w\mid w\in\mathbb{Z},\ w\geq 0\}\ \vee\ \mathbb{W}=\mathbb{N}\cup \{0\}$

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We can write the sets U, A and B in another way, but in set builder notation.

$\begin{aligned}&U=\{x\mid x=2^n,\ n\in\mathbb{W},\ n<8\}\\ \\ \Longrightarrow\ \ &U=\{x\mid x=2^n,\ n\in C=\{0,\ 1,\ 2,\ 3,\ 4,\ 5,\ 6,\ 7\}\}\end{aligned}$

$\begin{aligned}&A=\{y\mid y=4^n,\ n\in\mathbb{W},\ n<4\}\\ \\ \Longrightarrow\ \ &A=\{y\mid y=(2^2)^n,\ n\in\mathbb{W},\ n<4\}\\ \\ \Longrightarrow\ \ &A=\{y\mid y=2^{2n},\ n\in\mathbb{W},\ n<4\}\\ \\ \\ \textsf{Let}\ \ &2n=k.\ \ n\in\mathbb{W}\ \wedge\ n<4\ \Rightarrow\ n\in\{0,\ 1,\ 2,\ 3\}\ \Rightarrow\ k\in\{0,\ 2,\, 4,\ 6\}\\ \\ \\ \therefore\ \ \ \ &A=\{y\mid y=2^k,\ k\in D=\{0,\ 2,\ 4,\ 6\}\}\end{aligned}$

$\begin{aligned}&B=\{z\mid z=8^n,\ n\in\mathbb{W},\ n<2\}\\ \\ \Longrightarrow\ \ &B=\{z\mid z=(2^3)^n,\ n\in\mathbb{W},\ n<2\}\\ \\ \Longrightarrow\ \ &B=\{z\mid z=2^{3n},\ n\in\mathbb{W},\ n<2\}\\ \\ \\ \textsf{Let}\ \ &3n=m.\ \ n\in\mathbb{W}\ \wedge\ n<2\ \Rightarrow\ n\in\{0,\ 1\}\ \Rightarrow\ m\in\{0,\ 3\}\\ \\ \\ \therefore\ \ \ \ &B=\{z\mid z=2^m,\ m\in E=\{0,\ 3\}\}\end{aligned}$

Now,

$\begin{aligned}(i)\ \ &A'=U-A\\ \\ &A'=\{a\mid a=2^p,\ p\in (C-D)\}\\ \\ &A'=\{a\mid a=2^p,\ p\in\{1,\ 3,\ 5,\ 7\}\}\\ \\ &A'=\{a\mid a=2^p,\ p=2u-1,\ u\in\mathbb{N},\ u\leq 4\}\\ \\ \\ (ii)\ \ &B'=U-B\\ \\ &B'=\{b\mid b=2^q,\ q\in(C-E)\}\\ \\ &B'=\{b\mid b=2^q,\ q\in\{1,\ 2,\ 4,\ 5,\ 6,\ 7\}\}\\ \\ &B'=\{b\mid b=2^q,\ q\in\mathbb{N},\ q\leq 7,\ q\neq 3\}\end{aligned}$

$\begin{aligned}(iii)\ \ &(A\cap B)'=A'\cup B'\\ \\ &(A\cap B)'=\{c\mid c=2^r,\ r\in\{1,\ 2,\ 3,\ 4,\ 5,\ 6,\ 7\}\}\\ \\ &(A\cap B)'=\{c\mid c=2^r,\ r\in\mathbb{N},\ r\leq 7\}\end{aligned}$