Answer to Question #148090 in Discrete Mathematics for Promise Omiponle

Question

Determine the truth value of each of the quantified statements below if the domain consists of R.
(a)∃x(x^5 = -1)
(b)∃x(x^6< x^4)
(c)∀x((-x)4=x^4)
(d)∀x(2x > x)

Answer 1

(a) True. There is a x in R such that $x^5=-1$ , it's x=-1.

(b) True. $x^6<x^4,x^4(x^2-1)<0,x^4<0$ has no solutions so $x^2-1<0,|x|<1,-1<x<1$. So yes, there is x in R such that $x^6<x^4$ .

(c) True. No matter what x is, $(-x)^4=((-1)\cdot x)^4=(-1)^4\cdot x^4=1\cdot x^4=x^4$ .

(d) False. $2x>x, x>0$ . So this statement is true only for positive numbers, not for all x in R