#$&* course Mth 151 2/27/12 8:32PM 013. `query 13
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Given Solution: `a**The question asks for the truth values of p and q that would make the statement ~(p^q) false. If ~(p^q) is false then p^q is true, which means that both p and q must be true.** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q3.2.18 p false q true ~[(~p^~q) U ~q] YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: In (~p^~q), the ~q makes the whole statement false since ~q is false. ~q is also false in (~p^~q) U ~q making the whole thing false. So the truth value would be ~[(~p^~q) U ~q]. confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a**~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.** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q3.2.36 p: 15<8 q: 9 not > 5 r: 18 <= 18 evaluate -(p U -q) U -r YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: ~(p U ~q) and ~r are both false, and because it is a disjunction the whole statement is false. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** p and q are both false statements, while r is a true statement. It follows that p U ~q is true: since ~q is true the disjunction is true. It therefore follows that ~(p U ~q) is false. Since r is true, ~r is false. Thus ~(p U ~q) U ~r is a disjunction of two false statements, ~(p U ~q) and ~r. A disjunction of two false statements is false. So the statement is false. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q (formerly 3.2.42) This wasn't assigned, but you should be able to answer based on your responses to similar assigned questions. {}{}How many rows are there in a statement involving p,q,r,s,u,v,m,n? Note that rows go across the page. For example a statement involving just p and q will have four rows, one each for TT, TF, FT and FF. The headings (i.e., p, q and whatever other statements are necessary to evaluate the truth table) might also be considered a row, but for this problem do not consider the headings to be a row. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The number of possible values can be found by raising 2 to the number of statements power. So a truth table with 8 statements will 256 values and rows. 2^2 = 256 confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** 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. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q3.2.56 (fomerly 3.2.54) This was not assigned but based on your work on similar problems you should be able to construct the truth table for (-p ^ -q) U (~p U q). Give your truth table: YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The headings for the columns were: p, q, ~p, ~q, (~p ^ ~q), (~p U q), (~p ^ ~q) U (~p U q) First row: TTFFFTT Second row: TFFTFFF Third row: FTTFFTFF Fourth row: FFTTTTT confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** For column headings p q ~p ~q ~p^~q ~p U q (~p^~q) U (~p Uq) the first row would start off T T, for p and for q. Then F F for ~p and ~q. Then F for ~p ^ ~q, then T for ~p V q, then T for the final column. So the first row would be T T F F F T T. The second row would be T F F T F F F The third row would be F T T F F T T and the fourth row would be F F T T T T T ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q3.2.68 (formerly 3.2.66) This wasn't assigned but is similar to other assigned problems so you should be able to solve it: Negate the following statement using De Morgan's Law: ' F.C. tried to sell the wine but was unable to do so'. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: I translated the statement to ~p ^ ~q, and found the negate of it which was ~p U ~q. Next I changed the statement back to word form, so the statement would be “Fellman Chutz didn't try to sell the wind or he was able to do so.” confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** 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 ' F.C. tried to sell the book but was unable to do so' can be symbolized as 'p ^ q'. Its negation would be ~(p ^ q) = ~p U ~q. We translate this as 'F.C. didn't try to sell the book or he sold it', or something equivalent. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q3.2.68 (formerly 3.2.66) This wasn't assigned but is similar to other assigned problems so you should be able to solve it: Negate the following statement using De Morgan's Law: ' F.C. tried to sell the wine but was unable to do so'. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: I translated the statement to ~p ^ ~q, and found the negate of it which was ~p U ~q. Next I changed the statement back to word form, so the statement would be “Fellman Chutz didn't try to sell the wind or he was able to do so.” confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** We use two ideas here. The first is that 'but' is interpreted as 'and'; and the second is that the negation of an 'and' statement is an 'or' statement. deMorgan's Laws say that the negation of p OR q is ~p AND ~q, while the negation of p AND q is ~p OR ~q. The given statement ' F.C. tried to sell the book but was unable to do so' can be symbolized as 'p ^ q'. Its negation would be ~(p ^ q) = ~p U ~q. We translate this as 'F.C. didn't try to sell the book or he sold it', or something equivalent. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK #*&! ********************************************* Question: `q3.2.68 (formerly 3.2.66) This wasn't assigned but is similar to other assigned problems so you should be able to solve it: Negate the following statement using De Morgan's Law: ' F.C. tried to sell the wine but was unable to do so'. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: I translated the statement to ~p ^ ~q, and found the negate of it which was ~p U ~q. Next I changed the statement back to word form, so the statement would be “Fellman Chutz didn't try to sell the wind or he was able to do so.” confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** 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 ' F.C. tried to sell the book but was unable to do so' can be symbolized as 'p ^ q'. Its negation would be ~(p ^ q) = ~p U ~q. We translate this as 'F.C. didn't try to sell the book or he sold it', or something equivalent. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK #*&!#*&!