course Mth 151 阔ꊴּƇ{assignment #013
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14:51:51 `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'?
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RESPONSE --> all men are under six feet tall confidence assessment: 2
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14:54:47 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.
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RESPONSE --> Some men are not over six feet tall. self critique assessment: 2
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14:56:16 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.
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RESPONSE --> It is important to figure out the negation before you even think about which statement is true. self critique assessment: 2
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14:57:26 `q002. What is the negation of the statement 'some men are over six feet tall' ?
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RESPONSE --> Some men are not over six feet tall. All men are over six feet tall. confidence assessment: 2
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14:58:44 `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.
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RESPONSE --> No men is over six feet tall because there is an overlap between some men are over six feet tall and some men are not over six feet tall confidence assessment: 2
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14:59:27 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'.
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RESPONSE --> I must have missed the question to this answer somehow. self critique assessment: 2
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15:01:31 `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.
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RESPONSE --> 1. All roses are not black. 2. Some roses are black. 3. There was some dodo birds that was stupid. 4. Turtles are always slow. confidence assessment: 2
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15:03:06 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'.
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RESPONSE --> some roses are black all roses are black all Dodo birds were stupid some turtles were not slow self critique assessment: 2
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dȋRsU assignment #014 014. Truth Tables Liberal Arts Mathematics I 10-09-2008
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RESPONSE --> T T, T F, F T, F F confidence assessment: 2
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15:07:15 It is possible that p is true and q is true. Another possibility is that p is true and q is false. A third possibility is that p is false and q is true. A fourth possibility is that p is false and q is false. These possibilities can be listed as TT, TF, FT and FF, where it is understood that the first truth value is for p and the second for q.
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RESPONSE --> the first truth value is for p and the second for q self critique assessment: 2
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15:08:00 `q002. For each of the for possibilities TT, TF, FT and FF, what is the truth value of the compound statement p ^ q ?
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RESPONSE --> TT = T, TF = F, FT = F, FF = F confidence assessment: 2
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15:08:43 p ^ q means 'p and q', which is only true if both p and q are true. In the case TT, p is true and q is true so p ^ q is true. In the case TF, p is true and q is false so p ^ q is false. In the case FT, p is false and q is true so p ^ q is false. In the case FF, p is false and q is false so p ^ q is false.
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RESPONSE --> P^Q is p and q which is only true if both p and q is true self critique assessment: 2
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15:10:09 `q003. Write the results of the preceding problem in the form of a truth table.
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RESPONSE --> p and q p q p ^ q T T T T F F F T F F F F confidence assessment: 2
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15:10:41 The truth table must have headings for p, q and p ^ q. It must include a line for each of the possible combinations of truth values for p and q. The table is as follows: p q p ^ q T T T T F F F T F F F F.
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RESPONSE --> The truth table must have headings for p,q and p^q self critique assessment: 2
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15:11:31 `q004. For each of the possible combinations TT, TF, FT, FF, what is the truth value of the proposition p ^ ~q?
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RESPONSE --> TT = F TF = T FT = F FF = F confidence assessment: 2
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15:12:26 For TT we have p true, q true so ~q is false and p ^ ~q is false. For TF we have p true, q false so ~q is true and p ^ ~q is true. For FT we have p false, q true so ~q is false and p ^ ~q is false. For FF we have p false, q false so ~q is true and p ^ ~q is false.
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RESPONSE --> IF q is false then it would be true in this case vice versa. self critique assessment: 2
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15:13:40 `q005. Give the results of the preceding question in the form of a truth table.
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RESPONSE --> p ~p T F F T confidence assessment: 2
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15:14:48 The truth table will have to have headings for p, q, ~q and p ^ ~q. We therefore have the following: p q ~q p^~q T T F F T F T T F T F F F F T F
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RESPONSE --> THE TRUTH TABLE SHOULD HAVE TO HAVE THE HEADINGS FOR P,Q,~ Q AND P^ ~Q self critique assessment: 2
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15:16:11 `q006. Give the truth table for the proposition p U q, where U stands for disjunction.
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RESPONSE --> P OR Q P Q PUQ T T T T F T F T T F F F confidence assessment: 2
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15:16:50 p U q means 'p or q' and is true whenever at least one of the statements p, q is true. Therefore p U q is true in the cases TT, TF, FT, all of which have at least one 'true', and false in the case FF. The truth table therefore reads p q p U q T T T T F T F T T F F F
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RESPONSE --> PUQ MEANS P OR Q IT IS TRUE WHENEVER AT LEAST ONE OF THE STATEMENTS ARE TRUE self critique assessment: 2
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15:18:46 `q007. Reason out the truth values of the proposition ~(pU~q).
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RESPONSE --> IF P IS TRUE AND Q IS FALSE THEN (pU~q) = T U T = T and ~T = False confidence assessment: 2
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15:19:51 In the case TT p is true and q is true, so ~q is false. Thus p U ~q is true, since p is true. So ~(p U ~q) is false. In the case TF p is true and q is false, so ~q is true. Thus p U ~q is true, since p is true (as is q). So ~(p U ~q) is false. In the case FT p is false and q is true, so ~q is false. Thus p U ~q is false, since neither p nor ~q is true. So ~(p U ~q) is true. In the case FF p is false and q is false, so ~q is true. Thus p U ~q is true, since ~q is true. So ~(p U ~q) is false.
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RESPONSE --> ~ q is false. Thus p U ~q is false. self critique assessment: 2
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15:22:00 We need headings for p, q, ~q, p U ~q and ~(p U ~q). Our truth table therefore read as follows: p q ~q pU~q ~(pU~q) T T F T F T F T T F F T F F T F F T T F
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RESPONSE --> ~(pU~q) F F T F self critique assessment: 2
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