course MTH 151 8:42 PM 10/26/09 If your solution to stated problem does not match the given solution, you should self-critique per instructions at
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Given Solution: `a** This argument is an instance of the 'fallacy of the converse'. In commonsense terms we can say that there could be many reasons why he might be ecstatic--it doesn't necessarily follow that it's because he doesn't have to set up. A Venn diagram can be drawn with 'doesn't get up' inside 'ecstatic'. An x inside 'ecstatic' but outside 'doesn't get up' fulfills the premises but contradicts the conclusions. Also [ (p -> q) ^ q ] -> p if false for the p=F, q=F case. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Right I understand that it follows he could be ecstatic because it's Christmas, not necessarily only because he didn't have to set up. I wrote the argument as -p instead of p which is why I got that it was not valid if p is True and q is True ------------------------------------------------ Self-critique Rating:3 ********************************************* Question: `qQuery 3.6.11 (formerly 3.6.12). This wasn't assigned but you should be able to analyze it. {}{}She uses ecommerce or uses credit. She doesn't use credit. Therefore she uses ecommerce. {}{}Is the argument and valid or invalid and why? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Yes it is valid. It says explicitly that she must use one or the other. So if she doesn't use one, then she must use the other. confidence rating:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** The argument can be symbolized as p V q ~q therefore p This type of argument is called a disjunctive syllogism. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Which is a valid argument form. ------------------------------------------------ Self-critique Rating:3 ********************************************* Question: `qQuery 3.6.18 evaluate using the truth table: ~p -> q, p, therefore -q YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: [(-p -> q) ^ p] -> -q T T T F F T F F T F T T T T T F F F F T T F T T T T F F I found that the argument is invalid for ""p is True q is True"". confidence rating:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** We need to evaluate {(p--> ~q) ^ ~p} --> ~q, which is a compound statement representing the argument. p q ~p ~q (p--> ~q) {(p--> ~q) ^ ~p} {(p--> ~q) ^ ~p} --> ~q then truth table is p q ~p ~q (p--> ~q) {(p--> ~q) ^ ~p} {(p--> ~q) ^ ~p} --> ~q T T F F F F T T F F T T F T F T T F T T F F F T T T T T Note that any time p is true (p->~q)^~p) is false so the final conditional (p->~q)^p) -> ~q is true, and if q is false then ~q is true so the final conditional is true. The F in the third row makes the argument invalid. To be valid an argument must be true in all possible instances. } Another version of this problem has ~p -> q and p as premises, and ~q as the consequent. The headings for this version of the problem are: p q ~p ~q ~p -> q (~p -> q) ^ p [ (~p -> q) ^ p ] -> ~q. Truth values: T T F F T T F T F F T T T T F T T F F F T F F T T T F T The argument is not true by the final truth value in the first line. To be true the statement [ (~p -> q) ^ p ] -> ~q must be true for any set of truth values. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): The written question above did was not the same as the question answered in the Given Solution. I understand both statements. ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q3.6.24 evaluate using the truth table: ( (p ^ r) -> (r U q), and q ^ p), therefore r U p YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: p r q {[(p ^ r) -> (r U q)] ^ (q ^ p)]} -> (r U p) T T T T T T T T T T T T F T T T F F T T T F T F T T T T T T T F F F T F F F T T F F F F T F F F T F F T T F T T F F T T F T F F T T F F T T F F T F T T F F T F It is a valid statement. It is always true for all truth values of pqr confidence rating:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** The headings can be set up as follows: p q r p^r rUq (p^r)->(rUq) {((r ^ p ) --> (rU q)) ^ (q^p)} {((r ^ p ) --> (rU q)) ^ (q^p)} --> (rUp) This permits each column to be evaluated, once the columns for p, q and r are filled in by standard means, by looking at exactly two of the preceding columns. Here's the complete truth table. pqr r^p q^p rUp rUq (r^p)->(rUq) [(r^p)->(rUq)]^(q^p) {[(r^p) -> (rUq)] ^ q^p} -> rUp ttt t t t t t t t ttf f t t t t t t tft t f t t t f t tff f f t f t f t ftt f f t t t f t ftf f f f t t f t fft f f t t t f t fff f f f f t f t All T's in the last column show that the argument is valid. COMMON BAD IDEA: p, q, r, (r ^ p), (rUq), (q^p), (rUp), {[(r^p)->(rUq)] ^ (q^p)}->(rUp) You're much better off to include columns for [(r^p)->(rUq)] and {[(r^p)->(rUq)] ^ (q^p)} before you get to {[(r^p)->(rUq)] ^ (q^p)}->(rUp). If you have to look at more than two previous columns to evaluate the one you're working on you are much more likely to make a mistake, and in any case it takes much longer to evaluate. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK I understand. I find this all makes good sense. ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q3.6.30: Christina sings or Ricky isn't an idol. If Ricky isn't an idol then Britney doesn't win. Britney wins. Therefore Christina doesn't sing. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: No this is not a valid argument. I don't even have to make a truth table to figure this one out. Just because Britney wins, that doesn't make Ricky an idol. So there is no way to tell if Christina sings or not because we cannot find out if Ricky is an idol or not. All we know for sure is Britney won and that either Christina sings or Ricky isn't an idol. confidence rating:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** Solution using deductive reasoning: If r stands for RM is a teen idol c stands for CA sings b stands for BS wins then the statements are c U ~r ~r -> ~b b therefore ~c. The contrapositive of ~r -> ~b is b -> r. So we have b -> r b therefore r. We now have c U ~r r therefore c by disjunctive syllogism. That is, Britney wins so Rich is an idol. Christina sings or Ricky isn't an idol. So Christina sings. The argument concludes ~c, the Christina doesn't sing. So the argument is invalid Solution using truth tables: If we let p stand for Christina sings, r for Ricky Martin is a teen idol and w for Britney Spears wins AMA award then we have p V ~r ~r->~w w Therefore ~p The argument is the statement [(pV~r)^(~r->~w)^w]-~p We can evaluate this statement using the headings: p r w ~r ~w ~p (pV ~r) (~r->~w) [(pV~r)^(~r->~w)^w] [(pV~r)^(~r->~w)^w]-~p. We get p r w ~r ~w ~p (pV ~r) (~r->~w) [(pV~r)^(~r->~w)^w] [(pV~r)^(~r->~w)^w]-~p T T T F F F T T T F T T F F T F T T F T T F T T F F T F F T T F F T T F T T F T F T T F F T F T F T F T F F T T F F F T F F T T F T T T T T F F F T T T T T F T. The argument is not valid, being false in the case of the first row. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Right I understand. If all values are true then the statement is false. ------------------------------------------------ Self-critique Rating: ********************************************* Question: `qPrevious version 3.6.30 determine validity: all men are mortal. Socrates is a man. Therefore Socrates is mortal YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: It is a valid statement. confidence rating:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** This can be reasoned out by the transitive property of the conditional. If p stands for 'a man', q for 'mortal', r for 'Socrates' you have r -> p p -> q therefore r -> q which is valid by the transitive property of the conditional. A truth-table argument would evaluate [ (r -> p) ^ (p -> q) ] -> (r -> q). The final column would come out with all T's, proving the validity of the argument. ** STUDENT COMMENT I was saying p=men are mortal, q=Socrates is a man, and r= Socrates is mortal and so this is why I was having trouble. INSTRUCTOR RESPONSE p, q and r need to stand for simple statements. None of the statements you quote here is a simple statement. 'Socrates is a man' is not a simple statement. It means 'if it's Socrates, then it's a man'. Similarly 'Socrates is mortal' means 'if it's Socrates, then it's mortal'. The statement 'men are mortal' says 'if it's a man, then it's mortal'. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating:3 "