#$&* course Mth 151 3/16/2012 7:24PM 015. `query 15
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Given Solution: `a** 'Milk contains calcium' can be put into p -> q form as 'if it's milk then it contains calcium'. The converse of p -> q is q -> p, which would be 'if it contains calcium then it's milk' The inverse of p -> q is ~p -> ~q, which would be 'if it's not milk then it doesn't contain calcium'. The contrapositive of p -> q is ~q -> ~p, which would be 'if it doesn't contain calcium then it's not milk'. Note how the original statement and the contrapositive say the same thing, and how the inverse and the converse say the same thing. NOTE ON ANOTHER STATEMENT: If the statement is 'if it ain't broke don't fix it: Converse: If you don't fix it, then it ain't broke Inverse: If it's broke, then fix it. Contrapositive: If you fix it, then it's broke. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `qQuery 3.4.18 state the contrapositive of 'if the square of the natural number is even, then the natural number is even.' Using examples decide whether both are truth or false. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: P equals “if the square of a a natural number is even” and q equals “then the natural number is even”, and it is in p → q form. The contrapositive of statement would be “If the natural number is not even, then the square of the natural number is not even” because the contrapositive of p → q is ~q → ~p confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** The statement is of the form p -> q with p = 'square of nat number is even' and q = 'nat number is even'. The contrapositive of p -> q is ~q -> ~p, which in this case would read 'if a natural number isn't even then its square isn't even'. STUDENT RESPONSE WITH SOMEWHAT PICKY BUT IMPORTANT INSTRUCTOR CORRECTION: if the natural number isn't even , then the square of a natural numbewr isn't even Good. More precisely: if the natural number isn't even , then the square of THAT natural number isn't even. To say that the square of a natural number isn't even doesn't necessarily refer to the given uneven natural number. COMMON ERROR WITH INSTRUCTOR COMMENT: The natural number is not even, if the square of a natural number is not even. ex.-3^2=9,5^2=25 This statement is true. ** You have stated the inverse ~p -> ~q. It doesn't matter that the 'if' is in the second half of your sentence, the 'if' in your statement still goes with ~p when it should go with ~q. COMMON ERROR WITH INSTRUCTOR COMMENT: If the natural number is not even, then the square of the natural number is not even. This statement does not involve square roots. It addresses only squares. And 26 isn't the square of a natural number. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `qExplain how you used examples to determine whether both statements are true or both false. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: For the first statements I used the square of of 4 is 2 and the square of 16 is 4 and the square of 36 is 6 and so on and so forth as examples, and all of those examples were true. For the contrapositive of the statement I used the square of 9 is 3 and the square of 25 is 5 and the square of 49 is 7 and so on and so forth as examples, all of those are true as well. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** The first statement said that if the square of a natural number is even then the natural number is even. For example, 36 is the square of 6, 144 is the square of 12, 256 is the square of 16. These examples make us tend to believe that the statement is true. The contrapositive says that if the natural number is even then its square isn't even. For example, the square of the odd number 7 is 49, which is not an even number. The square of the odd number 13 is 169, which is not an even number. This and similar examples will convince us that this statement is true. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `qExplain why either both statements must be true, or both must be false. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Both statements are true because the statement and its contrapositive have the same truth table. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** The reason is that the truth tables for the statement and its contrapositive are identical, so if one is true the other is true and if one is false the other must be false. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `qQuery 3.4.24 write 'all whole numbers are integers' in form 'if p then q'. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The statement would be “if its a whole number then it is a integer”. P equals “its a whole number” and q equals “its an integer” confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** p could be 'it's a whole number' and q would then be 'it's an integer'. The statement would be 'if it's a whole number then it's an integer'. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `qQuery 3.4.30 same for ' principal hires more only if board approves YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The statement would be “if the principal hires more teachers, the school board approved”. P is “the principal hires more teachers” and q equals “the school board approved”. confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aCOMMON ERROR WITH INSTRUCTOR COMMENT: If the principal will hire more teachers, then the school board would approve. INSTRUCTOR COMMENT: p only if q is the same as if p then q; should be 'if the principle hires, the school board approved' ** STUDENT COMMENT I switched the two because I thought the 'only if' meant that was the p part. I thought that it made more sense that the teacher hiring was dependent on the board approving. INSTRUCTOR RESPONSE To say that 'the teacher hiring was dependent on the board approving' would be correct, and would have the same meaning as the instructor's stated solution. However the statement 'the teacher hiring was dependent on the board approving' is not equivalent to your statement 'If board approves then the principal hires more', which is not equivalent to the given statement. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `qQuery 3.4.48 true or false: 6 * 2 = 14 iff 9 + 7 neg= 16. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Both 6 * 2 = 14 and 9 + 7 not equal to 16 are false, the whole statement though is true because both parts are false. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** Both statments are false, but the compound statement is true. The compound statement 'p if and only if q' is equivalent to 'if p then q, AND if q then p'. This compound statement is true because p and q are both false, so 'if p then q' and 'if q then p' are both of form F -> F and therefore true ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `qQuery 3.4.55 contrary or consistent: ' this number is an integer. This number is irrational.' YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The statement would be contrary because an irrational number can not be an integer. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a**Any integer n can be expressed in the form p / q as n / 1. So all integers are rational. Irrational numbers are defined as those numbers which are not rational. So the statements are indeed contrary-it is impossible for a number to be both an integer and irrational. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK" Self-critique (if necessary): ------------------------------------------------ Self-critique rating: Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!