course MTH 151
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11:20:07 3.2.6 ~(p^q) false; truth values of components
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RESPONSE --> They must both be true, then the ~ is applied to make the statement false.
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11:20:17 **The question asks for the truth values of p and q that would make the statement ~(p^q) false. If ~(p^q) is false then p^q is true, which means that both p and q must be true.**
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RESPONSE --> right
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11:21:51 3.2.18 p false q true ~[(~p^~q) U ~q]
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RESPONSE --> (~p ^ ~ q) = T ^ F = F F V ~q = F V F = F ~F = T
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11:22:08 **~p ^ ~q is false because ~q is false. One false is fatal to a conjunction. ~q is false so both parts of the disjunction [(~p^~q) U ~q] are false. Thus [(~p^~q) U ~q] is false. The negation ~[(~p^~q) U ~q] of this statement is therefore true.**
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RESPONSE --> right
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11:25:00 3.2.36 p: 2>7 q: 8 not > 6 r: 19 <= 19 evaluate -(p U -q) U -r
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RESPONSE --> working within the paren first: (p V ~q) = F V T = T T V ~r = T V F = T ~T = F
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11:25:40 ** p and q are both false statements, while r is a true statement. It follows that p U ~q is true: since ~q is true the disjunction is true. It therefore follows that ~(p U ~q) is false. Since r is true, ~r is false. Thus ~(p U ~q) U ~r is a disjunction of two false statements, ~(p U ~q) and ~r. A disjunction of two false statements is false. So the statement is false. **
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RESPONSE --> ok
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11:26:48 3.2.42 How many rows are there in a statement involving p,q,r,s,u,v,m,n? Note that rows go across the page. For example a statement involving just p and q will have four rows, one each for TT, TF, FT and FF. The headings (i.e., p, q and whatever other statements are necessary to evaluate the truth table) might also be considered a row, but for this problem do not consider the headings to be a row.
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RESPONSE --> Using the formula we have 2^8 = 256
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11:27:11 ** If you just have two statements p and q, then there are four possible truth values: TT, TF, FT and FF. If you have three statements p, q and r then there are eight possible truth values: TTT, TTF, TFT, TFF, and FTT, FTF, FFT, FFF. Note that the number of possible truth values doubles every time you add a statement. The number of truth values for 2 statements is 4, which is 2^2. For 3 statements this doubles to 8, which is 2^3. Every added statement doubles the number, which adds a power to 2. From this we see that the number of possible truth values for n statements is 2^n. For the 8 statements listed for this problem, there are therefore 2^8 =256 possible truth values. **
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RESPONSE --> right
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11:32:15 3.2.54 truth table for (-p ^ -q) U (~p U q)
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RESPONSE --> Since I already had the truth values of each component from a previous problem, I did a kind of shorthand method here where I worked out each statement's truth value in my head then entered it into the table. (~p ^ ~ q) V (~p V q) F V T = T F V F = F F V T = T T V T = T
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11:34:14 ** For column headings p q ~p ~q ~p^~q ~p U q (~p^~q) U (~p Uq) the first row would start off T T, for p and for q. Then F F for ~p and ~q. Then F for ~p ^ ~q, then T for ~p V q, then T for the final column. So the first row would be T T F F F T T. The second row would be T F F T F F F The third row would be F T T F F T T and the fourth row would be F F T T T T T **
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RESPONSE --> I checked these with my tables from #'s 51 and 54 and they are correct.
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11:35:17 3.2.66 negate using De Morgan's Law: ' V.M. tried to sell the book but she was unable to do so'.
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RESPONSE --> Pauline Mula did not try to sell the book or she was able to do so.
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11:35:47 ** We use two ideas here. The first is that 'but' is interpreted as 'and'; and the second is that the negation of an 'and' statement is an 'or' statement. deMorgan's Laws say that the negation of p OR q is ~p AND ~q, while the negation of p AND q is ~p OR ~q. The given statement ' V.M. tried to sell the book but she was unable to do so' can be symbolized as 'p ^ q'. Its negation would be ~(p ^ q) = ~p U ~q. We translate this as 'V.M. didn't try to sell the book or she sold it', or something equivalent. **
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RESPONSE --> right
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11:37:03 3.2.78 is the statement 3 + 1 = 4 xor 2 + 5 = 9 true or false?
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RESPONSE --> true, because one statement is true (3+1 = 4) but the other is false.
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11:37:14 ** For an XOR statement exactly one part has to be true. The statement is true because the first part is true and the second is false. We need exactly one true statement; if both parts were true the XOR wouldn't be. **
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RESPONSE --> right
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