State whether the following statements are true or false and justify your answers. (a) ∃x in Z ∃y in Z (x + y = 2) ∧ (x - y = 3)

listgrein6u

listgrein6u

Answered question

2022-09-04

State whether the following statements are true or false and justify your answers
(a) x Z y Z ( x + y = 2 ) ( x y = 3 )
(b) x R y R x > y x 2 < y 2
(c) x R y R x 2 + y 2 > 1

Answer & Explanation

Bordenauaa

Bordenauaa

Beginner2022-09-05Added 18 answers

Step 1
a) If we operate in the second condition we get x = y + 3; which, substituting in the first one, gives us: ( 3 + y ) + y = 2. Or what is the same y = 1 / 2, so we can not choose y to be an integer.
Step 2
b) Just take x and y bigger or equal than 1, e.g. x = 2 and y = 1 would do the trick.
Step 2
c) Take x R . If x > 1 or x < 1 then x 2 > 1, and we can take y = 0 R . If 1 x 1, in particular x 2 < 2, so 2 x 2 > 0. We can then take y = 2 x 2 R and get y 2 + x 2 = 2 > 1.
We see that in this case, the statement is true, unlike the other ones.

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