course Mth 151
.................................................
......!!!!!!!!...................................
19:59:29 `q001. Note that there are 4 questions in this assignment. Suppose I tell you 'If it rains today, I'll give you $100.' Under which of the following circumstances can you claim that I was not telling the truth? 1. It rains and I give you $100. 2. It rains and I don't give you $100. 3. It doesn't rain and I give you $100. 4. I doesn't rain and I don't give you $100.
......!!!!!!!!...................................
RESPONSE --> 2. It rains and I don't give you $100
.................................................
......!!!!!!!!...................................
20:00:01 I said what would happen under a certain condition. In situation #2, that condition is fulfilled and what I said would happen doesn't happen. Therefore in situation #2 it is clear that I wasn't telling the truth. In situation #3, the condition that I addressed isn't fulfilled so no matter what happens I can't be accused of not telling the truth. I said what would happen if rains. No matter what happens, if it doesn't rain what I said cannot be held against me. It should be clear to anybody that situation #1 is exactly what you would expect, and that situation #4 is just would you would probably expect from my statement in the event that it doesn't rain, so nobody would say that this situation violates my claim to truthfulness.
......!!!!!!!!...................................
RESPONSE --> ok
.................................................
......!!!!!!!!...................................
20:01:56 `q002. Suppose that tell you 'It will rain today and I will give you $100'. Under which of the following circumstances can you claim that I was not telling the truth? 1. It rains and I give you $100. 2. It rains and I don't give you $100. 3. It doesn't rain and I give you $100. 4. I doesn't rain and I don't give you $100.
......!!!!!!!!...................................
RESPONSE --> 3. It doesn't rain and I give you $100.
.................................................
......!!!!!!!!...................................
20:02:56 It should be clear that situation #1 completely fulfills the conditions of my statement. Both of the things that I say will happen do happen. In situation #2, it rains but you don't get the $100. I said two things were going to happen and one of them didn't. In that case you would have to say that I wasn't telling truth. In situation #3, again one of the things I say is going to happen does but the other doesn't, so again you would have to say that I wasn't telling truth. In situation #4, neither of the things I say will happen does and certainly it would have to be said that I wasn't telling truth.
......!!!!!!!!...................................
RESPONSE --> ok
.................................................
......!!!!!!!!...................................
20:12:08 `q003. Suppose that tell you 'It will rain today or I will give you $100, but not both'. Under which of the following circumstances can you claim that I was not telling the truth? 1. It rains and I give you $100. 2. It rains and I don't give you $100. 3. It doesn't rain and I give you $100. 4. I doesn't rain and I don't give you $100.
......!!!!!!!!...................................
RESPONSE --> #1. both happened so the truth was not told #4. Neither of the two happened and at least one was supposed to happen, so the truth was not told.
.................................................
......!!!!!!!!...................................
20:12:23 In situations 2 and 3, one of the things happens and the other doesn't, so you would not be able to say that I wasn't telling the truth. However in situation 1, both things happen, which I said wouldn't be the case; and in situation 4 neither thing happens. In both of these situations you would have to say that I was not telling truth.
......!!!!!!!!...................................
RESPONSE --> ok
.................................................
......!!!!!!!!...................................
20:15:57 `q004. Suppose that tell you 'It will rain today or I will give you $100'. Under which of the following circumstances can you claim that I was not telling the truth? 1. It rains and I give you $100. 2. It rains and I don't give you $100. 3. It doesn't rain and I give you $100. 4. I doesn't rain and I don't give you $100.
......!!!!!!!!...................................
RESPONSE --> #4. if it doesn't rain then i get $100. Well in #4 it doesn't rain and i don't get $100, so the truth was not told.
.................................................
......!!!!!!!!...................................
20:16:23 At first this might seem to be the same as the preceding problem. But in the preceding problem we specifically said '... but not both.' In this case that qualification was not made. Therefore we have regard the statement as true as long as at least one of the conditions is fulfilled. This is certainly the case for situation 1: both conditions are true we can certainly say that at least one is true. So in situation #1 we have to regard the present statement as true. So situation #1 would not be included among those in which I could be accused of not telling the truth.
......!!!!!!!!...................................
RESPONSE --> ok
.................................................
z噮l~xū Student Name: assignment #013
.................................................
......!!!!!!!!...................................
20:31:45 `q001. There are 4 questions in this set. Two statements are said to be negations of one another if exactly one of the statements must be true. This means that if one statement is true the other must be false, and if one statement is false the other must be true. What statement is the negation of the statement 'all men are over six feet tall'?
......!!!!!!!!...................................
RESPONSE --> No men are over six feet tall
.................................................
......!!!!!!!!...................................
20:32:32 You might think that the negation would be 'no men are over six feet tall'. However, the negation is in fact 'some men are not over 6 feet tall'. The negation of a statement, in addition to being false whenever the statement is true, has to include every possibility except those covered by the statement itself. With respect to men being over six feet tall, there are three possibilities: 1. All men are over six feet tall, 2. no men are over six feet tall, and 3. some men are over six feet tall while others aren't. It should be clear that statements 1 and 2 do not cover the possibility of the third. In fact no two of these statements cover the possibility of the remaining one. However the following two statements do cover all possibilities: All men are over six feet tall (the original statement), and some men are not over six feet tall. The second statement might seem to be identical to statement 3, 'some men are over six feet tall while others aren't', but it is not. The statement 'some men are not over six feet tall' does not address whether there are men over six feet tall or not, while statement 3 states that there are. And the statement 'some men are not over six feet tall' might seem to leave out the possibility of statement 2, 'no men are over six feet tall', but again it doesn't address whether or not there are also men over six the tall. Therefore the negation of the statement 'all men are over six feet tall' is 'some men are not over six feet tall'.
......!!!!!!!!...................................
RESPONSE --> ok
.................................................
......!!!!!!!!...................................
20:33:28 It doesn't matter what's true and what isn't. If the question was to write the negation of 'all men are under 20 feet tall' you would still state the negation as 'some men are under 20 feet tall'. In this case the negation is true, which proves that the statement itself is false.
......!!!!!!!!...................................
RESPONSE --> ok
.................................................
......!!!!!!!!...................................
20:34:07 In the given problem the negation 'some men are under 6 ft tall' is true, proving that the original statement 'all men are over 6 ft tall' is false.
......!!!!!!!!...................................
RESPONSE --> ok
.................................................
......!!!!!!!!...................................
20:34:24 These examples demonstrate why it is important to figure out the negation before you even thing about which statement is true. Either the statement or its negation will be true, but never both.
......!!!!!!!!...................................
RESPONSE --> ok
.................................................
......!!!!!!!!...................................
20:35:27 `q002. What is the negation of the statement 'some men are over six feet tall' ?
......!!!!!!!!...................................
RESPONSE --> no men are over 6 feet tall
.................................................
......!!!!!!!!...................................
20:36:13 While it might seem that the negation of this statement is 'some men are not over six feet tall', the correct negation is 'no men are over six feet tall'. This is because there is an 'overlap' between 'some men are over six feet tall' and 'some men are not over six feet tall' because both statements are true if some men are over six feet while some are under six feet. Negations have to be exact opposites--if one statement is true the other must be false--in addition to the condition that the two statements cover every possible occurance. Again we have the three possibilities, 1. All men are over six feet tall, 2. no men are over six feet tall, and 3. some men are over six feet tall while others aren't. The statement ' some men are over six feet tall' is consistent with statements 1 and 3, because if all men are over six feet tall then certainly some men are over 6 feet tall, and if some men are over 6 feet tall and others aren't, it is certainly true that some men are over six feet tall. The only statement not consistent with 'some men are over six feet tall' is Statement 2, 'No men are over six feet tall'. Thus this statement is the negation we are looking for.
......!!!!!!!!...................................
RESPONSE --> ok
.................................................
......!!!!!!!!...................................
20:45:44 `q003. As seen in the preceding two questions, the negation of a statement that says 'all are' or 'all do' is 'some aren't' or 'some don't', and the negation of a statement that says 'some are' or 'some do' is 'all aren't' or 'none are', or 'all do not' or 'none do'. Each of the following statements can be expressed as and 'all' statement or a 'some' statement. Identify which is which and give the negation of each statement: 1. Every dog has its day. 2. Some roses are black. 3. Every attempt fails. 4. In some cases the desired outcome isn't attained.
......!!!!!!!!...................................
RESPONSE --> 1. some dogs don't have its day = all statement 2. no roses are black = some statement 3. some attempts fail = all statement 4. in all cases the desired outcome isn't attained = some statement
.................................................
......!!!!!!!!...................................
20:46:43 Statement 1 can be expressed as 'All dogs do have their day', a form of 'all do'. The negation of 'all do' is 'some don't'. In this case the negation might be expressed as 'some dogs do not have their day'. Statement 2 is a straightforward 'some are' statement having negation 'all are not', expressed in this case as 'no roses are black', or equivalently 'there are no black roses'. Statement 3 can be restated equivalently in 'all do' form as 'all attempts do fail', and is negated in 'some don't' form as 'some attempts do not fail', or equivalently as 'some attempts succeed'. Statement 4 can be equivalently expressed in 'some are' form as 'some outcomes are not as desired'. This statement is negated by the 'none are' form as 'no outcomes are not as desired', which can then be expressed as 'all outcomes are as desired'.
......!!!!!!!!...................................
RESPONSE --> ok
.................................................
......!!!!!!!!...................................
20:51:39 `q004. Negate the following statements: 1. No roses are black. 2. Some roses are not black. 3. There were Dodo birds that weren't stupid. 4. There were never turtles that weren't slow.
......!!!!!!!!...................................
RESPONSE --> 1. some roses are black 2. some roses are black 3. there was not dudo birds that weren't stupid 4. there are some turtles that weren't slow
.................................................
......!!!!!!!!...................................
20:52:22 Statement 1 says that there is no such thing as a rose which is not black, which says that all roses fail to be black. The negation of 'all are' is 'some aren't', so the negation of 'all roses are not black' is 'some roses are not not black', which is the same as 'some roses are black'. Statement 2 is a 'some are' statement, negated in the 'all are not' form by 'all roses are not not black', or equivalently, 'all roses are black'. Statement 3 is equivalent to saying that 'some Dodos birds were not stupid', negated as 'all are not' in the form 'all Dodo birds were not not stupid', or equivalently as 'all Dodo birds were stupid'. Statement 4 is equivalent of saying that 'all turtles were slow', equivalent of the 'all are' form. This is negated in 'some are not' form by 'some turtles were not slow'.
......!!!!!!!!...................................
RESPONSE --> ok
.................................................
ݱΕǗRiz assignment #013 űpZ Liberal Arts Mathematics I 02-26-2006
......!!!!!!!!...................................
20:18:54 3.2.6 ~(p^q) false; truth values of components
......!!!!!!!!...................................
RESPONSE --> both components would be true.
.................................................
......!!!!!!!!...................................
20:19:03 **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.**
......!!!!!!!!...................................
RESPONSE --> ok
.................................................
......!!!!!!!!...................................
20:19:37 3.2.18 p false q true ~[(~p^~q) U ~q]
......!!!!!!!!...................................
RESPONSE --> true
.................................................
......!!!!!!!!...................................
20:19:42 **~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.**
......!!!!!!!!...................................
RESPONSE --> ok
.................................................
......!!!!!!!!...................................
20:20:27 3.2.36 p: 2>7 q: 8 not > 6 r: 19 <= 19 evaluate -(p U -q) U -r
......!!!!!!!!...................................
RESPONSE --> false
.................................................
......!!!!!!!!...................................
20:20:34 ** 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. **
......!!!!!!!!...................................
RESPONSE --> ok
.................................................
......!!!!!!!!...................................
20:21:52 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.
......!!!!!!!!...................................
RESPONSE --> 2^8= 256 rows
.................................................
......!!!!!!!!...................................
20:21:59 ** 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. **
......!!!!!!!!...................................
RESPONSE --> ok
.................................................
......!!!!!!!!...................................
20:22:44 3.2.54 truth table for (-p ^ -q) U (~p U q)
......!!!!!!!!...................................
RESPONSE --> T, F, T, T
.................................................
......!!!!!!!!...................................
20:24:30 ** 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 **
......!!!!!!!!...................................
RESPONSE --> ok, I had me chart just like the answers, but I just listed the answers for the last row, top to bottom.
.................................................
......!!!!!!!!...................................
20:25:57 3.2.66 negate using De Morgan's Law: ' V.M. tried to sell the book but she was unable to do so'.
......!!!!!!!!...................................
RESPONSE --> Pauline Mula did not try to sell the book or she was able to do so.
.................................................
......!!!!!!!!...................................
20:27:28 ** 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. **
......!!!!!!!!...................................
RESPONSE --> ok, mine was wored different, but I think its the same answer
.................................................
......!!!!!!!!...................................
20:28:49 3.2.78 is the statement 3 + 1 = 4 xor 2 + 5 = 9 true or false?
......!!!!!!!!...................................
RESPONSE --> True, because the first is true and the second is false
.................................................
......!!!!!!!!...................................
20:29:00 ** 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. **
......!!!!!!!!...................................
RESPONSE --> ok
.................................................
"