Question

Find the truth set of each of these predicates where the domain is the set of integers.

(a) \(P\left( x \right):{x^3} > 1\)

(b) \(Q\left( x \right):{x^2} = 2\)

(c) \(R\left( x \right):x < {x^2}\)

Short answer

(a) \(\left\{ {1,2,3,....} \right\}\)

(b) \(\phi \)

(c) \(\left\{ {..... - 3, - 2, - 1,2,3,4......} \right\}\)

Step-by-step solution

Truth set

The truth set contains all integers for which the given equation is true.

To find the truth set of the given predicate (a) 

The truth set of \(P\left( x \right):{x^3} > 1\)consist of all integers x for which the inequality \({x^3} > 1\) is true.

Hence, all positive integers are in the truth set of \(P\left( x \right):{x^3} > 1\)

Thus, truth set for \(P\left( x \right):{x^3} > 1\) is \(\left\{ {1,2,3,....} \right\}\)

To find the truth set of the given predicate (b) 

The truth set of \(Q\left( x \right):{x^2} = 2\)consist of all integers x for which the equality \({x^2} = 2\) is true.

Whereas, the solutions of \({x^2} = 2\)are \(x = - \sqrt 2 \) and \(x = \sqrt 2 \) which are not integers.

Thus, truth set for \(Q\left( x \right):{x^2} = 2\) is empty \(\phi \) .

To find the truth set of the given predicate (c) 

The truth set of \(R\left( x \right):x < {x^2}\)consist of all integers x for which the inequality \(x < {x^2}\) is true.

If x is an integer , then \(x < {x^2}\)is positive and all positive integers are greater than negative integer.

Hence, the truth set contains all positive integers.

Now, since \({0^2} = 0\) , 0 is not an element of the truth set.

And since \({1^2} = 1\) , 1 is not an element of the truth set.

If \(x \ge 2\), then \(x < {x^2}\) and thus all positive integers of 2 are in the truth set.

Thus, truth set for \(R\left( x \right):x < {x^2}\) is \(\left\{ {..... - 3, - 2, - 1,2,3,4......} \right\}\) .