91Ó°ÊÓ

Consider the following statement: \(\forall\) basketball players \(x, x\) is tall.

Let \(D=E=\\{-2,-1,0,1,2\\}\). Write negations for each of the following statements and determine which is true, the given statement or its negation. a. \(\forall x\) in \(D, \exists y\) in \(E\) such that \(x+y=1\). b. \(\exists x\) in \(D\) such that \(\forall y\) in \(E, x+y=-y\).

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. For all students \(x\), if \(x\) studies discrete mathematics, then \(x\) is good at logic. Tarik studies discrete mathematics. \- Tarik is good at logic.

Consider the following statement: $$ \exists x \in \mathbf{R} \text { such that } x^{2}=2 \text {. } $$

Let \(D=\\{-48,-14,-8,0,1,3,16,23,26,32,36\\}\). Determine which of the following statements are true and which are false. Provide counterexamples for those state ments that are false. a. \(\forall x \in D\), if \(x\) is odd then \(x>0\). b. \(\forall x \in D\), if \(x\) is less than 0 then \(x\) is even. c. \(\forall x \in D\), if \(x\) is even then \(x \leq 0\). d. \(\forall x \in D\), if the ones digit of \(x\) is 2 , then the tens digit is 3 or 4 . e. \(\forall x \in D\), if the ones digit of \(x\) is 6 , then the tens digit is 1 or 2 .

In each of 14-19, (a) rewrite the statement in English without using the symbol \(\forall\) or \(\exists\) but expressing your answer as simply as possible, and (b) write a negation for the statement. $$ \forall r \in \mathbf{Q} . \exists \text { integers } a \text { and } b \text { such that } r=a / b \text {. } $$

Let \(D\) be the set of all students at your school, and let \(M(s)\) be "s is a math major," let \(C(s)\) be " \(s\) is a computer science student," and let \(E(s)\) be " \(s\) is an engineering student." Express each of the following statements using quantifiers, variables, and the predicates \(M(s), C(s)\), and \(E(s)\). a. There is an engineering student who is a math major. b. Every computer science student is an engineering student. c. No computer science students are engineering students. d. Some computer science students are also math majors. e. Some computer science students are engineering students and some are not.

Consider the following statement: \(\forall\) integers \(n\), if \(n^{2}\) is even then \(n\) is even. Which of the following are equivalent ways of expressing this statement? a. All integers have even squares and are even. b. Given any integer whose square is even, that integer is itself even. c. For all integers, there are some whose square is even. d. Any integer with an even square is even. e. If the square of an integer is even, then that integer is even. f. All even integers have even squares.

\(\forall n \in \mathbf{Z}\), if \(n\) is prime then \(n\) is odd or \(n=2\).

Rewrite each of the following statements in the form \(" v\) \(x\), if then \(x\) and \(y\), if then \("\) or " a. All Java programs have at least 5 lines. b. Any valid argument with true premises has a true conclusion. c. The sum of any two even integers is even. d. The product of any two odd integers is odd.