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Show that each of these conditional statements is a tautology by using truth tables. a ) [ n o t p ? ? ( p
Show that each of these conditional statements is a tautology by using truth tables.
a
b
c
d
Let be the statement has a cat," let be the statement has a dog," and let be the statement has a ferret." Express each of these statements in terms of quantifiers, and logical connectives.
Let the domain consist of all students in your class.
a A student in your class has a cat, a dog, and a ferret.
b All students in your class have a cat, a dog, or a ferret.
c Some student in your class has a cat and a ferret, but not a dog.
d No student in your class has a cat, a dog, and a ferret.
e For each of the three animals, cats, dogs, and ferrets, there is a student in your class who has this animal as a pet.
Translate each of these nested quantifications into an English statement that expresses a mathematical fact. The domain in each case consists of all real numbers.
a
b
c
d
Use rules of inference to show that the hypotheses If it does not rain or if it is not foggy, then the sailing race will be held and the lifesaving demonstration will go onIf the sailing race is held, then the trophy will be awarded," and "The trophy was not awarded" imply the conclusion It rained."
Prove that if is an integer and is even, then is even using
a a proof by contraposition.
b a proof by contradiction.
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