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Question

In each of the following, decide whether the given quantified statement is true or false (the domain for both  and  is the set of all real numbers). Provide a brief justification in each case.

1. 1.)             
2. 2.) 

Answer 1

2) is true. Let y = 1, Then, for all real numbers x, x + y > xy since when y = 1 we get x + 1 > x.

1) is false. Pick any real number x. Then for y = 1 - x, we get x + y = 1. For example, if we try x = 2, then let y = 1 - 2 = -1 to get x + y = 1. So, for any x we can find a real number y so that x + y = 1.

Answer 2

1)

To negate it it is enough to find one example, where the statement is not true (counterexample). We take a real number x and see if for at least one y x+y ≠1 violates

Let's set x =1, then for y=0 the statement doesn't hold. So 1) is not always True.x+y=1+0=1.

Similar for 2)