Please help with the following question!!!

Question

Which of the following quantified statements is (are) true? Let the domain for  be the set of all real numbers.

[Note: There may be multiple correct answers!]

a.) 

b.)

c.)

d.)  

e.)

f.)

Answer 1

Fizaa, here we go!

a) False - x² = 0 if x is = 0 (it's true for all other x ≠ 0)

b) True - e^x > 0 for all real numbers, x (although e^x ≠ 0 for any real x, the ≥ sign in the question is still true)

c) False - x² ≥ 0 for all real x

d) True - see part b) above

e) True - there is such an x and it is = +1

f) False - this is not true for all x, only for x = 1; see part e) above

Answer 2

a) must be true since x² is greater than 0 for all x that are real

b) must be true since exp(x) is positive for all real values of x.

c) is not true. There is no real number whose square is negative. side note, this is why complex numbers were developed.

d) is true. lim x→ -∞ exp(x) = 0, but x never really equals 0. It only gets infinitely close to 0 and stays positive.

e) is true. x=1 solves x+1=2x. There is at least one solution to the equation.

f) is false. only x=1 solves x+1=2x. if we plug in x=2 we get 3=4 which is impossible. So not all values of x satisfy the equation.