Question

1. Formalize the following argument by using the given predicates and then rewriting the argument as a numbered sequence of statements. Identify each statement as either a premise, or a conclusion that follows according to a rule of inference from previous statements. In that case, state the rule of inference and refer by number to the previous statements that the rule of inference used.

Lions hunt antelopes. Ramses is a lion. Ramses does not hunt Sylvester. Therefore, Sylvester is not an antelope.

Predicates: \(\mathrm{H}(\mathrm{x}, \mathrm{y})=" \mathrm{x}\) hunts \(\mathrm{y} ", \mathrm{~L}(\mathrm{x})=" \mathrm{x}\) is a lion" and \(\mathrm{A}(\mathrm{x})=" \mathrm{x}\) is an

antelope". The domain of discourse is all animals.

2. Prove that there can be no perfect square between 25 and 36 , i.e. there is no integer \(n\) so that \(25<n^{2}<36\). Prove this by directly proving the negation.

Your proof must only use integers, inequalities and elementary logic. You may use that inequalities are preserved by adding a number on both sides, or by multiplying both sides by a positive number. You cannot use the square root function. Do not write a proof by contradiction.

3. Prove that for any positive integer \(n\), there is an even positive integer \(k\) so that

$$ \frac{1}{n+2} \leq \frac{1}{k-1}<\frac{1}{n} $$

4. Prove by contraposition for arbitrary \(x \neq-2\) : if \(x\) is irrational, then so is \(\frac{x}{x+2}\).

Answer 1

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