Problem 8 Prove or give a counterexample to each of the foll

Problem 8: Prove or give a counterexample to each of the following statements. (8) Vx E R.3y E R, (x + y > 0 or x + y = 0) (9) Yx E R.ZyER, (x + y > 0 and x + y = 0) (10) (Vx ER.ZyER, x + y > 0) and (viz E R.3y E R, x + y = 0)

Solution

Ans(1):

we know that adding opposite of a number to itself will always give 0 and to get greater than 0 value we can add any positive integer so choosing y=-x+1 will always satisfy the given statement

x+y=x-x+1=1 which is clearly >0

Hence it is true for all cases.

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Ans(2):

we know that subtracting a number from itself will always give 0 and to get greater than 0 value we can add any positive integer so choosing y=x-1 will always satisfy the given statement

x-y=x-(x-1)=x-x+1=1 which is clearly >0

hence it is True.

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Ans(3):

It is False because for some given x, x+y will not always be >0 for all y in R

like say x=2 and y=-5

then x+y=2-5=-3 which is clearly not >0

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Ans(4):

it will fail for x=0

as we know product of 0 with any real number is again 0

while condition xy>0 doesn\'t include 0

hence it is False