#$&* course Mth 151 05/06/20111206 If your solution to stated problem does not match the given solution, you should self-critique per instructions at
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Given Solution: `a** 'Milk contains calcium' can be put into p -> q form as 'if it's milk then it contains calcium'. The converse of p -> q is q -> p, which would be 'if it contains calcium then it's milk' The inverse of p -> q is ~p -> ~q, which would be 'if it's not milk then it doesn't contain calcium'. The contrapositive of p -> q is ~q -> ~p, which would be 'if it doesn't contain calcium then it's not milk'. Note how the original statement and the contrapositive say the same thing, and how the inverse and the converse say the same thing. NOTE ON ANOTHER STATEMENT: If the statement is 'if it ain't broke don't fix it: Converse: If you don't fix it, then it ain't broke Inverse: If it's broke, then fix it. Contrapositive: If you fix it, then it's broke. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): N/A ------------------------------------------------ Self-critique Rating:3 ********************************************* Question: `qQuery 3.4.18 state the contrapositive of 'if the square of the natural number is even, then the natural number is even.' Using examples decide whether both are truth or false. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: WOULD THE CONTRAPOSITIVE READ: IF THE SQUARE OF A NATURAL NUMBER IS NOT EVEN, THEN THE NATURAL NUMBER IS NOT EVEN confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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): I USED AN IF, THAN STATEMENT. I DON’T THINK THIS IS CORRECT. READING THROUGH THE THIS GIVEN SOLUTION HAS HELPED A LOT WITH UNDERSTNADING HOW TO LOOK AT THIS PROBLEM. ------------------------------------------------ Self-critique Rating:3 ********************************************* Question: `qExplain how you used examples to determine whether both statements are true or both false. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: FOR THE STATEMENT: IF THE SQUARE OF A NATURAL NUMBER IS EVEN,M THEN THE NATURAL NUMBER IS EVEN. THIS COULD ME THAT 2*2=4 OR 4*4=16 OR 12*12=144. FOR THE CONTRAPOSITIVE PORTION, TO DISPROVE THIS, WE COULD DO 3*3=9. THIS IS NOT EVEN, THEREFORE VOIDS THE TRUE STATEMENT. confidence rating #$&*:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** The first statement said that if the square of a natural number is even then the natural number is even. For example, 36 is the square of 6, 144 is the square of 12, 256 is the square of 16. These examples make us tend to believe that the statement is true. The contrapositive says that if the natural number is even then its square isn't even. For example, the square of the odd number 7 is 49, which is not an even number. The square of the odd number 13 is 169, which is not an even number. This and similar examples will convince us that this statement is true. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): N/A ------------------------------------------------ Self-critique Rating: ********************************************* Question: `qExplain why either both statements must be true, or both must be false. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: BOTH STATEMENTS WOULD HAVE TO BE TRUE BECAUSE THERE IS NOT A NUMBER OUT THERE THAT IS A NATURAL NUMBER WHEN IT IS SQUARED DOES NOT CHANGE FROM EVEN TO ODD. EVENS STICK WITH EVENS, AND ODDS STICK WITH ODDS. THIS WOULD BE THE SAME FOR THE FALSE STATEMENT AS WELL confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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): THAT’S WHAT I SAID BUT IN A MUCH MORE DRAWN OUT WAY ------------------------------------------------ Self-critique Rating:3 ********************************************* Question: `qQuery 3.4.24 write 'all whole numbers are integers' in form 'if p then q'. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: IF ITS AN WHOLE NUMBER, THEN IT IS AN INTEGER confidence rating #$&*:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** p could be 'it's a whole number' and q would then be 'it's an integer'. The statement would be 'if it's a whole number then it's an integer'. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):N/A ------------------------------------------------ Self-critique Rating:3 ********************************************* Question: `qQuery 3.4.30 same for ' principal hires more only if board approves YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: IF THE PRINCIPAL WILL HIRE MORE TEACHERS, THEN THE SCHOOL BOARD WOULD APPROVE confidence rating #$&*:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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): N/A ------------------------------------------------ Self-critique Rating:3 ********************************************* Question: `qQuery 3.4.48 true or false: 6 * 2 = 14 iff 9 + 7 neg= 16. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: THIS IS FALSE. 6*2=12 AND 9 +7=16. THIS IS THE OPPOSITE OF WHAT IT IS SAYING. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** Both statments are false, but the compound statement is true. The compound statement 'p if and only if q' is equivalent to 'if p then q, AND if q then p'. This compound statement is true because p and q are both false, so 'if p then q' and 'if q then p' are both of form F -> F and therefore true ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OKAY. I CAN SEE WHY IT IS TRUE. BC OF BOTH STATEMENTS BEING FALSE ------------------------------------------------ Self-critique Rating: ********************************************* Question: `qQuery 3.4.55 contrary or consistent: ' this number is an integer. This number is irrational.' YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: THIS IS CONTRARY. YOU CANNOT HAVE BOTH AN INTEGER AND AN IRRATIONAL NUMBER AT THE SAME TIME. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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): N/A ------------------------------------------------ Self-critique Rating:3 " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!