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14\. At the start of a program (written in pseudocode) the integer variable \(n\) is assigned the value \(7 .\) Determine the value of \(n\) after each of the following successive statements is encountered during the execution of this program. [Here the value of \(n\) following the execution of the statement in part (a) becomes the value of \(n\) for the statement in part (b), and so on, through the statement in part (d). For positive integers \(a, b,\lfloor a / b\rfloor\) returns the integer part of the quotient - for example, \(\lfloor 6 / 2\rfloor=3\), \(\lfloor 7 / 2\rfloor=3,\lfloor 2 / 5\rfloor=0\), and \([8 / 3\rfloor=2 .]\) a) if \(n>5\) then \(n:=n+2\) b) if \(((n+2=8)\) or \((n-3=6)\) ) then \(n:=2 * n+1\) c) if \(((n-3=16)\) and \((\lfloor n / 6\rfloor=1)\) ) then \(n:=n+3\) d) if \((n \neq 21)\) and \((n-7=15)\) ) then \(n:=n-4\)

15\. 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) \(p \leftrightarrow q\)

16\. Prove that for every integer \(n, n^{2}\) is even if and only if \(n\) is even.

19\. For each of the following statements state the converse, inverse, and contrapositive. Also determine the truth value for each given statement, as well as the truth values for its converse, inverse, and contrapositive. (Here "divides" means "exactly divides.") a) [The universe comprises all positive integers.] If \(m>n\), then \(m^{2}>n^{2}\) b) [The universe comprises all integers.] If \(a>b\), then \(a^{2}>b^{2}\). c) [The universe comprises all integers.] If \(m\) divides \(n\) and \(n\) divides \(p\), then \(m\) divides \(p\). d) [The universe consists of all real numbers.] \(\forall x\left[(x>3) \rightarrow\left(x^{2}>9\right)\right]\) e) [The universe consists of all real numbers.] For all real numbers \(x\), if \(x^{2}+4 x-21>0\), then \(x>3\) or \(x<-7\)

21\. 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) \(\exists x \exists y[(2 x+y=5) \wedge(x-3 y=-8)]\) e) \(\exists x \exists y[(3 x-y=7) \wedge(2 x+4 y=3)]\)

21\. Prove that for all real numbers \(x\) and \(y\), if \(x+y \geq 100\), then \(x \geq 50\) or \(y \geq 50\)

24\. Let \(n\) be an integer. Prove that \(n\) is even if and only if \(31 n+12\) is even.

25\. Let the universe for the variables in the following statements consist of all real numbers. In each case negate and simplify the given statement. a) \(\forall x \forall y[(x>y) \rightarrow(x-y>0)]\) b) \(\forall x \forall y[(x