Chapter 1: Q11E (page 108)
Prove that there exists a pair of consecutive integers such that one of these integers is a perfect square and the other is a perfect cube.
The pair of consecutive integral values 8 and 9, such that 8 is a perfect cube and 9 is a perfect square.
And, the proof is constructive.
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Show that is a tautology
Let p and q be the propositions
p: I bought a lottery ticket this week.
q: I won the million-dollar jackpot.
Express each of these propositions as an English sentence.
a)
b)
c)
d)
e)
f )
g)
h)
When does , where s is a compound proposition
Write each of these statements in the form “if p, then q” in English. [Hint: Refer to the list of common ways to express conditional statements provided in this section.]
a) It is necessary to wash the boss’s car to get promoted.
b) Winds from the south imply a spring thaw.
c) A sufficient condition for the warranty to be good is that you bought the computer less than a year ago.
d) Willy gets caught whenever he cheats.
e) You can access the website only if you pay a subscription fee.
f ) Getting elected follows from knowing the right people.
g) Carol gets seasick whenever she is on a boat.
Express these system specifications using the propositions p "The user enters a valid password," q "Access is granted," and r "The user has paid the subscription fee" and logical connectives (including negations).
a) "The user has paid the subscription fee, but does not enter a valid password."
b) "Access is granted whenever the user has paid the subscription fee and enters a valid password."
c) "Access is denied if the user has not paid the subscription fee."
d) "If the user has not entered a valid password but has paid the subscription fee, then access is granted."
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