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

 

 

Question:  `qQuery 3.4.6 write converse, inverse, contrapositive of ' milk contains calcium'

 

 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

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):

 

 

 

 

 

 

 

 

 

 

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.

 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

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):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

Question:  `qExplain how you used examples to determine whether both statements are true or both false.

 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

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):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

Question:  `qExplain why either both statements must be true, or both must be false.

 

 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

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):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

Question:  `qQuery 3.4.24 write 'all whole numbers are integers' in form 'if p then q'.

 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

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):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

Question:  `qQuery 3.4.30 same for ' principal hires more only if board approves

 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

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):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

Question:  `qQuery 3.4.48 true or false:  6 * 2 = 14 iff 9 + 7 neg= 16.

 

 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

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):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

Question:  `qQuery 3.4.55 contrary or consistent: ' this number is an integer.  This number is irrational.'

 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

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):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating: