91Ó°ÊÓ

Indicate which of the following statements are true and which are false. Justify your answers as best as you can. a. Every integer is a real number. b. 0 is a positive real number. c. For all real numbers \(r_{+}-r\) is a negative real number. d. Every real number is an integer.

Which of the following is a negation for "All dogs are loyal"? More than one answer may be correct. a. All dogs are disloyal. b. No dogs are loyal. c. Some dogs are disloyal. d. Some dogs are loyal. e. There is a disloyal animal that is not a dog. f. There is a dog that is disloyal. g. No animals that are not dogs are loyal. \(\mathrm{h}\). Some animals that are not dogs are loyal.

The following statement is true: " \(\forall\) nonzero numbers \(x, \exists\) a real number \(y\) such that \(x y=1 . .\) For each \(x\) given below, find a \(y\) to make the predicate " \(x y=1\) " true, a. \(x=2\) b. \(x=-1\) c. \(x=3 / 4\)

Let \(P(x)\) be the predicate " \(x>1 / x\)." a. Write \(P(2), P\left(\frac{1}{2}\right), P(-1), P\left(-\frac{1}{2}\right)\), and \(P(-8)\), and indicate which of these statements are true and which are false. b. Find the truth set of \(P(x)\) if the domain of \(x\) is \(\mathbf{R}\), the set of all real numbers. c. If the domain is the set \(\mathbf{R}^{+}\)of all positive real numbers, what is the truth set of \(P(x)\) ?

Use universal instantiation or universal modus ponens to fill in valid conclusions for the arguments in \(2-4 .\) \(\forall\) real numbers \(r, a\), and \(b\), if \(r\) is positive, then \(\left(r^{a}\right)^{b}=r^{a b}\). \(r=3, a=1 / 2\), and \(b=6\) are particular real numbers such that \(r\) is positive.

Write an informal negation for each of the following statements: a. All pots have lids. b. All birds can fly. c. Some pigs can fly, d. Some dogs have spots.

Consider the statement "There are no simple solutions to life's problems." Write an informal negation for the statement, and then write the statement formally using quantifiers and variables.

Some of the arguments in 7-18 are valid by universal modus ponens or universal modus tollens; others are invalid and exhibit the converse or the inverse error. State which are valid and which are invalid. Justify your answers. All healthy pcople cat an apple a day. Herbert is not a healthy person. Herbert does not eat an apple a day.

Statement: The sum of any two irrational numbers is irrational. Proposed negation: The sum of any two irrational numbers is rational.

Some of the arguments in 7-18 are valid by universal modus ponens or universal modus tollens; others are invalid and exhibit the converse or the inverse error. State which are valid and which are invalid. Justify your answers. All cheaters sit in the back row. Monty sits in the back row. \- Monty is a cheater.