- #1
docnet
Gold Member
- 799
- 484
- Homework Statement
- Consider the following statements about real numbers x and y. If true, prove it. If false, state its negation and prove that statement.
- Relevant Equations
- 1. ##\forall x>0, \exists y>0## such that ##xy<1##.
2. ##\exists y>0## such that ##\forall x>0, xy<1##.
1. True Let ##y=\frac{1}{x+1}##. Then ##xy=\frac{x}{x+1}<1, \forall x>0##.
2. False. Let ##x=\frac{1}{y}##. Then for any ##x>0##, you always have ##xy=\frac{y}{y}=1##
I feel like the second statement is wrong because I am choosing any ##y## , instead of choosing any ##x##.
2. False. Let ##x=\frac{1}{y}##. Then for any ##x>0##, you always have ##xy=\frac{y}{y}=1##
I feel like the second statement is wrong because I am choosing any ##y## , instead of choosing any ##x##.
Last edited: