91Ó°ÊÓ

In 21 and 22 , rewrite each statement without using variables or the symbol \(\forall\) or \(\exists\). Indicate whether the statement is true or false. a. \(\forall\) nonzero real numbers \(r, \exists\) a real number \(s\) such that \(r s=1\). b. \(\exists\) a real number \(s\) such that \(\forall\) real numbers \(r, r s=1\).

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

Rewrite each of the following statements in the two forms \(" \forall x\), if then \("\) and " \(x\), (without an if-then). a. The square of any even integer is even. b. Every computer science student needs to take data structures.

If an integer is divisible by 2 , then it is even.

Indicate whether the arguments in \(21-26\) are valid or invalid. Support your answers by drawing diagrams. No vegetarians eat meat. All vegans are vegetarian. No vegans eat meat.

Consider the statement "All integers are rational numbers but some rational numbers are not integers." a. Write this statement in the form " \(\forall x\), if then b. Let Ratl \((x)\) be \(x\) such that

If the square of an integer is odd, then the integer is odd.

Let the domain of \(x\) be the set \(\mathbf{Z}\) of integers, and let \(\operatorname{Odd}(x)\) be " \(x\) is odd," Prime \((x)\) be " \(x\) is prime," and Square \((x)\) be " \(x\) is a perfect square." (An integer \(n\) is said to be a perfect square if, and only if, it equals the square of some integer. For example, 25 is a perfect square because \(25=5^{2}\).) a. \(\exists x\) such that Prime \((x) \wedge \sim \operatorname{Odd}(x)\). b. \(\forall x, \operatorname{Prime}(x) \rightarrow \sim \operatorname{Square}(x) .\) c. \(\exists x\) such that \(\operatorname{Odd}(x) \wedge \operatorname{Square}(x)\).

In exercises \(28-33\), reorder the premises in each of the arguments to show that the conclusion follows as a valid consequence from the premises. It may be helpful to rewrite the statements in ifthen form and replace some statements by their contrapositives. Exercises 28-30 refer to the kinds of Tarski worlds discussed in Example 2.1.12 and 2.3.1. Exercises 31 and 32 are adapted from Symbolic Logic by Lewis Carroll.* 1\. All the objects that are to the right of all the triangles are above all the circles. 2\. If an object is not above all the black objects, then it is not a square. 3\. All the objects that are above all the black objects are to the right of all the triangles. \(\therefore\) All the squares are above all the circles.

In 32-38, (a) rewrite the statement formally using quantifiers and variables, and (b) write a negation for the statement. $$ \text { Everybody loves somebody. } $$