Write down the truth set of each of the following open sentences:

(i) $p (x) : x + 5 < 9, x \in N .$

(ii) $p (x) : x + 3 < 3, x \in N .$

(iii) $p (x) : x + 5 > 7, x \in R .$

(iv) $P (x) : 2 x^{2} + 5 x - 3 = 0, x \in 1$

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Solution

We have:

(i) Truth Set = ${x \in N : x + 5 < 9}$

${x \in N : x < 4} = {1, 2, 3}$

(ii) Truth Set = ${x \in N : x + 3 < 3}$

${x \in N : x < 0} = Φ .$

(iii) Truth Set = ${x \in R : x + 5 > 7}$

${x \in R : x > 2} =] 2, \infty [$

(iv) $2 x^{2} + 5 x - 3 = 0 \Rightarrow 2 x^{2} + 6 x - x - 3 = 0 \Rightarrow 2 x (x + 3) - (x + 3) = 0$

$\Rightarrow (x + 3) (2 x - 1) = 0 \Rightarrow x = - 3 o r x = \frac{1}{2} .$

$∴$ truth set = ${x \in l : 2 x^{2} + 5 x - 3 = 0} = {- 3} [∵ \frac{1}{2} \notin l]$

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