91Ó°ÊÓ

Write the negation of each of the following statements as an English sentence- without symbolic notation. (Here the universe consists of all the students at the university where Professor Lenhart teaches.) a) Every student in Professor Lenhart's Pascal class is majoring in computer science or mathematics. b) At least one student in Professor Lenhart's Pascal class is a history major. c) A student in Professor Lenhart's Pascal class has read all of her research papers on data structures.

For primitive statements \(p, q\), a) verify that \(p \rightarrow[q \rightarrow(p \wedge q)]\) is a tautology. b) verify that \((p \vee q) \rightarrow[q \rightarrow q]\) is a tautology by using the result from part (a) along with the substitution rules and the laws of logic. c) is \((p \vee q) \rightarrow[q \rightarrow(p \wedge q)]\) a tautology?

Write each of the following arguments in symbolic form. Then either establish the validity of the argument or provide a counterexample to show that it is invalid. a) If it is cool this Friday, then Craig will wear his suede jacket if the pockets are mended. The forecast for Friday calls for cool weather, but the pockets have not been mended. Therefore Craig won't be wearing his suede jacket this Friday. b) The contract will be fulfilled if and only if the new windows are installed in the house in June. If the new windows are installed in June, then Evelyn can move into her new house on the first of July. If she can't move in on July 1 , then Evelyn must pay the July rent on her apartment. The windows have been installed in June or Evelyn must pay the July rent on her apartment. Therefore Evelyn won't have to pay rent on her apartment for July.

Define the connective "Nand" or "Not ... and..." by \((p \uparrow q) \Leftrightarrow \neg(p \wedge q)\), for any statements \(p, q\). Represent the following using only this connective. a) \(\neg p\) b) \(p \vee q\) c) \(p \wedge q\) d) \(p \rightarrow q\) e) \(p \leftrightarrow q\)

Write the negation of each of the following true statements. For parts (a), (b), and (c) the universe consists of all integers; for parts (d) and (e) the universe comprises all real numbers. a) For all integers \(n\), if \(n\) is not (exactly) divisible by 2 , then \(n\) is odd. b) If the square of an integer is odd, then the integer is odd. c) If \(k, m, n\) are any integers where \(k-m\) and \(m-n\) are odd, then \(k-n\) is even. d) If \(x\) is a real number where \(x^{2}>16\), then \(x<-4\) or \(x>4\). e) For all real numbers \(x\), if \(|x-3|<7\), then \(-4

Let \(m, n\) be two positive integers. Prove that if \(m, n\) are perfect squares, then the product \(m n\) is also a perfect square.

\text { Prove that for all real numbers } x \text { and } y \text {, if } x+y \geq 100 \text {, then } x \geq 50 \text { or } y \geq 50 \text {. }

For the following statements the universe comprises all nonzero integers. Determine the truth value of each statement. a) \(\exists x \exists y[x y=1]\) b) \(\exists x \forall y[x y=1]\) c) \(\forall x \exists y[x y=1]\) d) \(\forall x \forall y\left[\sin ^{2} x+\cos ^{2} x=\sin ^{2} y+\cos ^{2} y\right]\) e) \(\exists x \exists y[(2 x+y=5) \wedge(x-3 y=-8)]\) f) \(\exists x \exists y[(3 x-y=7) \wedge(2 x+4 y=3)]\)

\text { In the arithmetic of real numbers, there is a real number, namely } 0 \text {, called the identity of }addition because \(a+0=0+a=a\) for every real number \(a\). This may be expressed in symbolic form by $$ \exists z \forall a[a+z=z+a=a] \text {. } $$ (We agree that the universe comprises all real numbers.) a) In conjunction with the existence of an additive identity is the existence of additive inverses. Write a quantified statement that expresses "Every real number has an additive inverse." (Do not use the minus sign anywhere in your statement.) b) Write a quantified statement dealing with the existence of a multiplicative identity for the arithmetic of real numbers. c) Write a quantified statement covering the existence of multiplicative inverses for the nonzero real numbers. (Do not use the exponent \(-1\) anywhere in your statement.) d) Do the results in parts (b) and (c) change in any way when the universe is restricted to the integers?

Consider the quantified statement \(\forall x \exists y[x+y=17]\). Determine whether this statement is true or false for each of the following universes: (a) the integers; (b) the positive integers; (c) the integers for \(x\), the positive integers for \(y\); (d) the positive integers for \(x\), the integers for \(y\).