Question

b) If P is the solution set of -3x + 4 < 2x - 3, XEN, and is the solution set of4x - 5 < 12, X€W. Find () PnQ (ii) Q - P​

Answer 1

$\large\underline\blue{\bold{Given \ :-  }}$

$\begin{gathered}\begin{gathered}\bf\begin{cases} &\sf{P \: is \: solution \: set \: of \: -3x + 4 < 2x - 3,x \: \epsilon \: {N}} \\ &\sf{Q \: is \: solution \: set \: of \: 4x - 5 < 12 \: x \: \epsilon \: W} \end{cases}\end{gathered}\end{gathered}$

$\large\underline\blue{\bold{To \: Find :-  }}$

$\begin{gathered}\begin{gathered}\bf \begin{cases} &\sf{P \: ∩ \: Q} \\ &\sf{P \: - Q} \end{cases}\end{gathered}\end{gathered}$

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$\large\underline\purple{\bold{Solution :- }}$

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$\bf \:❥︎ \: \underline{ Step :- 1.}$

$\sf \:  ⟼ -3x + 4 < 2x - 3, \: x \in \: N$

$\sf \:  ⟼ - 3x - 2x < - 3 - 4$

$\sf \:  ⟼ - 5x < - 7$

$\sf \:  ⟼x > \dfrac{7}{5}$

❥︎ It implies, x = 2, 3, 4, ... as x is natural number

$\bf\implies \:P = \{ x : x = n, \: n \in \: N,n \geqslant 2\}$

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$\bf \:❥︎ \: \underline{ Step :- 2.}$

$\sf \:  ⟼4x - 5 < 12 \: when \: x \in \: W$

$\sf \:  ⟼4x < 12 + 5$

$\sf \:  ⟼4x < 17$

$\sf \:  ⟼x < \dfrac{17}{4}$

❥︎ it implies, x = 0, 1, 2, 3, 4 as x is whole number

$\bf\implies \:Q = \{ 0,1,2,3,4 \}$

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$\bf \:❥︎ \: \underline{ Step :- 3.}$

$\sf \:  ⟼ \: ❥︎ \: To \: find \: P ∩ Q$

$\bf\ \:P = \{ x : x = n, \: n \in \: N,n \geqslant 2\}$

❥︎ and

$\bf\ \:Q = \{ 0,1,2,3,4 \}$

$\bf\:So, \: P \cap Q = \{ 2,3,4\}$

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$\bf \:❥︎ \: \underline{ Step :- 4.}$

$\bf \:  ⟼ ❥︎ \: To \: find \: P - Q$

$\bf \:P = \{ x : x = n, \: n \in \: N,n \geqslant 2\}$

❥︎ and

$\bf \:Q = \{ 0,1,2,3,4 \}$

$\bf\implies \:P - Q= \{ x : x = n, \: n \in \: N,n \geqslant 5\}$

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