course Mth 151
If your solution to stated problem does not match the given solution, you should self-critique per instructions at
http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm
Your solution, attempt at solution. If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.
015. `query 15
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Self-critique (if necessary):
Self-critique Rating:
Question: **** `qQuery 3.4.6 write converse, inverse, contrapositive of ' milk contains calcium'
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Your solution:
A) If it contains calcium, then it is milk.
B) If it is not milk, then it does not contain calcium.
C) If it does not contain calcium, then it is not milk.
Confidence Assessment:
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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. **
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Self-critique (if necessary):
Do I need to include the p? and q??
It's always a good idea to include the symbols to be sure you have interpreted the expressions properly; however as long as you give the correct statements your answers are correct.
Self-critique Rating:
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.
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Your solution:
Original: If the square of the natural number is even, then the natural number is even.
Contrapositive: If the natural number is not even, then the square of the natural number is not even.
Confidence Assessment:
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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. **
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Self-critique (if necessary):
So am I wrong or not? I keep forgetting the p? and q?.
Your answer is correct and well-stated.
Self-critique Rating:
Question: **** `qExplain how you used examples to determine whether both statements are true or both false.
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Your solution:
I am not sure what you are asking?
Confidence Assessment:
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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. **
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Self-critique (if necessary):
I wasn? thinking that way when I did them. I was looking at my paper that has conditional statements on it such as p->q (if p, then q.), for inverse ~p->~q (if not p, then not q) and so on.
That's right, but this question asks about the meaning of the statement, not just the logical equivalence of the two statements.
Self-critique Rating:
Question: **** `qExplain why either both statements must be true, or both must be false.
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Your solution:
I don? know why I know it is that a way.
Confidence Assessment:
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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. **
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Self-critique (if necessary):
I basically knew that just couldn? word it.
Self-critique Rating:
Question: **** `qQuery 3.4.24 write 'all whole numbers are integers' in form 'if p then q'.
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Your solution:
p=whole numbers
q=integers
p->q
If it is a whole number, then it is a integer.
Confidence Assessment:
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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'. **
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Self-critique (if necessary):
Self-critique Rating:
Question: **** `qQuery 3.4.30 same for ' principal hires more only if board approves
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Your solution:
p=principle will hire teachers
q=school board approves
p->q
If the principle hires more teachers, then the school board approved.
Confidence Assessment:
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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' **
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Self-critique (if necessary):
Self-critique Rating:
Question: **** `qQuery 3.4.48 true or false: 6 * 2 = 14 iff 9 + 7 neg= 16.
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Your solution:
this is false.
Confidence Assessment:
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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 **
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Self-critique (if necessary):
I understand now.
Self-critique Rating:
Question: **** `qQuery 3.4.55 contrary or consistent: ' this number is an integer. This number is irrational.'
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Your solution:
Contrary
Confidence Assessment:
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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. **
This looks good. See my notes. Let me know if you have any questions.