#$&* course MTH 151 11:27 Am 3/2/2015 012. `query 12
.............................................
Given Solution: `a** A sentence is a statement if it is true or false. Otherwise it is not a statement. 'There goes a big one' is not a statement because the word 'big' is open to interpretation so is not a statement. 'There are 3.87 * 10^89 particles in the universe at this instant' is a statement: it is either true or it isn't, though we don't know enough to tell which. The gross receipts for a film can be regarded as hard facts--unlike opinions on whether a film is good, or artistic. If 'a top grossing film' is defined as, say, a top-10 film in gross receipts, then we could ascertain whether it is true or false and we would have a the statement. However, 'top grossing' isn't defined here--does it mean one of the top three for the week, top 10 for the year, or what?--and for that reason we can't decide for sure whether it is true or false. So this sentence couldn't be regarded as a statement. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique Rating:3 ********************************************* Question: `qQuery Not assigned, but you should be able to answer: Is 'Sit up and behave.' a statementlineCount = lineCount + 1: bLine$(lineCount) = "" YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Its a command not a statement. confidence rating #$&*:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: `a** This is not a statement. It is a command. You could evaluate the truth of the statement 'you sat up', but not the truth of the command to sit up. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique Rating:3 ********************************************* Question: `qQuery 3.1.30 negate 'some people have all the luck YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: NOt all people have all the luck. confidence rating #$&*:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: `a** The negation would be 'all people don't have all the luck', which means 'nobody has all the luck'. The negation of 'some do' is indeed 'all do not', which is the same as 'none do'. The negation of 'all do' is 'some do not'. The negation of 'none do' is 'some do'. COMMON ERROR: Not everyone has all the luck, or equivalently some people do not have all the luck. This is not incompatible with the original statement, and the negation must be incompatible. Both would be true if some do have all the luck and some don't. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok
.............................................
Given Solution: `a** The statement is 'She has green eyes or he is 56 yrs. old' ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique Rating:3 ********************************************* Question: `qQuery 3.1.48 What is the statement -(p disjunction q) YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: She doesn't have green eyes and he is 56 years old. confidence rating #$&*:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: `a** The correct translation is 'It is not the case that she has green eyes or he is 56 yrs. old'. An equivalent statement, using deMorgan's Laws, would be 'she doesn't have green eyes and he is 56 years old' COMMON ERROR: She doesn't have green eyes or he is not 56 years old. This statement negates p V q as ~p V ~q, which is not correct. The negation of p V q is ~p ^ ~q. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique Rating:3 ********************************************* Question: `qQuery 3.1.54 Jack is an English Major or Chris collects DVDs, and it is not the case that both are so YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: (p U q) ^ ~(p ^ q) confidence rating #$&*:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: `a** The statement 'jack English or Chris collects' is symbolized by (p U q). The statement that it is not the case that both are so is symbolized ~(p ^ q). The entire statement is therefore (p U q) ^ ~(p ^ q).** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique Rating:3 ********************************************* Question: `q3.1.68 (formerly 3.1.60). This was not assigned, but you should be able to reason this out: {}{}True or false: there exists an integer that is not a rational number. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: False all are rational numbers. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: `a** A rational number is a number that can be written as p / q, with p and q both integers. Examples are 2/3, -5489/732, 6/2, etc.. Other examples could be 5/1, 12/1, -26/1; these of course reduce to just 5, 12, and -26. The point is that any integer can be written in this form, with 1 in the denominator, so any integer is in fact also a rational number. Thus there is no integer that is not a rational number, and the statement is false. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique Rating:3 ********************************************* Question: `qQuery 3.1.74 (was 3.1.66) Not assigned, but reason it out: True or false: each rat number is a positive number. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: FAlse not all rat numbers are positive. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: `a** This is false, and to prove it you need only give an example of a rational number that is negative. For example, -39/12 is a rational number (integer / integer) and is negative. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique Rating:3 ********************************************* Question: `qQuery 3.1.75 difference between 'all students did not pass the test' is the statement ' not all students passed the test' YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: NOne of the students passed and at least one of the students passed. confidence rating #$&*:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: `a** The first statement says that 'all students did not pass', meaning every student didn't pass, i.e., nobody passed the test. The second statement says that not everyone passed--at least one student didn't pass. The second statement doesn't address the question of whether anyone passed or not, so it doesn't necessarily say that some students did pass, but it leaves open the possibility that some did. Since the second statement contains possibilities the first does not the statements are not equivalent. ** " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `qQuery 3.1.75 difference between 'all students did not pass the test' is the statement ' not all students passed the test' YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: NOne of the students passed and at least one of the students passed. confidence rating #$&*:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: `a** The first statement says that 'all students did not pass', meaning every student didn't pass, i.e., nobody passed the test. The second statement says that not everyone passed--at least one student didn't pass. The second statement doesn't address the question of whether anyone passed or not, so it doesn't necessarily say that some students did pass, but it leaves open the possibility that some did. Since the second statement contains possibilities the first does not the statements are not equivalent. ** " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!
#$&* course MTH 151 3/2/2015 If your solution to stated problem does not match the given solution, you should self-critique per instructions at
.............................................
Given Solution: `a**The question asks for the truth values of p and q that would make the statement ~(p^q) false. If ~(p^q) is false then p^q is true, which means that both p and q must be true.** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique Rating:3 ********************************************* Question: `q3.2.18 p false q true ~[(~p^~q) U ~q] YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: True confidence rating #$&*:2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
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:3 ********************************************* Question: `q3.2.36 p: 15<8 q: 9 not > 5 r: 18 <= 18 evaluate -(p U -q) U -r YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: False confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: `a** p and q are both false statements, while r is a true statement. It follows that p U ~q is true: since ~q is true the disjunction is true. It therefore follows that ~(p U ~q) is false. Since r is true, ~r is false. Thus ~(p U ~q) U ~r is a disjunction of two false statements, ~(p U ~q) and ~r. A disjunction of two false statements is false. So the statement is false. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique Rating:3 ********************************************* Question: `q (formerly 3.2.42) This wasn't assigned, but you should be able to answer based on your responses to similar assigned questions. {}{}How many rows are there in a statement involving p,q,r,s,u,v,m,n? Note that rows go across the page. For example a statement involving just p and q will have four rows, one each for TT, TF, FT and FF. The headings (i.e., p, q and whatever other statements are necessary to evaluate the truth table) might also be considered a row, but for this problem do not consider the headings to be a row. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 256 possible confidence rating #$&*:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
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:3 ********************************************* 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: T T F F F T T. T F F T F F F F T T F F T T F F T T T T T confidence rating #$&*:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: `a** For column headings p q ~p ~q ~p^~q ~p U q (~p^~q) U (~p Uq) the first row would start off T T, for p and for q. Then F F for ~p and ~q. Then F for ~p ^ ~q, then T for ~p V q, then T for the final column. So the first row would be T T F F F T T. The second row would be T F F T F F F The third row would be F T T F F T T and the fourth row would be F F T T T T T ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique Rating:3 ********************************************* Question: `q3.2.68 (formerly 3.2.66) This wasn't assigned but is similar to other assigned problems so you should be able to solve it: Negate the following statement using De Morgan's Law: ' F.C. tried to sell the wine but was unable to do so'. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: F.C did not sell the wine or was able to do so. confidence rating #$&*:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
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:3 ********************************************* Question: `q3.2.81 is the statement 3 + 1 = 6 xor 2 + 5 = 7 true or false? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: It is true because one is false and the other is true. confidence rating #$&*:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: `a** For an XOR statement exactly one part has to be true. The statement is true because the second part is true and the first is false. We need exactly one true statement; if both parts were true the XOR wouldn't be. ** " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `q3.2.81 is the statement 3 + 1 = 6 xor 2 + 5 = 7 true or false? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: It is true because one is false and the other is true. confidence rating #$&*:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: `a** For an XOR statement exactly one part has to be true. The statement is true because the second part is true and the first is false. We need exactly one true statement; if both parts were true the XOR wouldn't be. ** " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!