Find the argument form for the following argument and determine whether it is valid. Can we conclude that the conclusion is true if the premises are true?
If George does not have eight legs, then he is not a spider. George is a spider.
∴ George has eight legs.
What rule of inference is used in each of these arguments?
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(a) Kangaroos live in Australia and are marsupials. Therefore, kangaroos are mar- supials.
(b) It is either hotter than 100 degrees today or the pollution is dangerous. It is less than 100 degrees outside today. Therefore, the pollution is dangerous.
(c) Linda is an excellent swimmer. If Linda is an excellent swimmer, then she can work as a lifeguard. Therefore, Linda can work as a lifeguard.
(d) Steve will work at a computer company this summer. Therefore, this summer Steve will work at a computer company or he will be beach bum.
(e) If I work all night on this homework, then I can answer all the exercises. If I answer all the exercises, I will understand the material. Therefore, if I work all night on this homework, then I will understand the material.
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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 on,” “If the sailing race is held, then the trophy will be awarded,” and “The trophy was not awarded” imply the conclusion “It rained.”
For each of these arguments, explain which rules of inference are used for each step.
(a) “Each of five roommates, Melissa, Aaron, Ralph, Veneesha, and Keeshawn, has taken a course in discrete mathematics. Every student who has taken a course in discrete mathematics can take a course in algorithms. Therefore, all five room- mates can take a course in algorithms next year.”
(b) “All movies produced by are wonderful. produced a movie about coal miners. Therefore, there is a wonderful movie about coal miners.”
Determine whether these are valid arguments
(a) If x is a positive real number, then x2 is a positive real number. Therefore, if
a2 is positive, where a is a real number, then a is a positive real number.
(b)Ifx2 ̸=0,wherexisarealnumber,thenx̸=0. Letabearealnumberwith
a2 ̸=0;thena̸=0. Problem 6
Use a direct proof to show that the product of two odd numbers is odd.
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Prove that if n is a perfect square, then n + 2 is not a perfect square.
Prove that if m and n are integers and mn is even, then m is even or n is even.
What is wrong with this famous supposed “proof” that 1 = 2?
“Proof”: We use these steps, a and b are two positive integers.
Steps ans reasons
1. a=bGiven
2. a2 = ab Multiplying both sides by a
3. a2 − b2 = ab − b2 Subtract b2 from both sides 4. (a−b)(a+b)=b(a−b)Factorbothsides
5. a+b=bDividebothsidesbya−b
6. 2b=bReplaceabybsincea−b
7. 2=1Dividebothsidesbyb
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