Open Query 3-2

#$&*

course MTH 151

Time of submission: 5:46 PM, 19 Feb 2012

If your solution to stated problem does not match the given solution, you should self-critique per instructions athttp://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.

013. `query 13

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Question: `q3.2.6 ~(p^q) false; truth values of components

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Your solution:

- P and q must both be true. Because the ~ negates the original validity, and the statement is thus false.

confidence rating #$&*: 3

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Given Solution:

`a**The question asks for the truth values of p and q that would make the statement ~(p^q) false. If ~(p^q) is false then p^q is true, which means that both p and q must be true.**

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Self-critique (if necessary):

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Question: `q3.2.18 p false q true ~[(~p^~q) U ~q]

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Your solution:

- Substitute p and q for their respective truth values:

= ~[( (F) ^ ~ (T) ) V ~ (T) ]

= ~[ (T ^ F ) V F ]

= ( F ^ T ) V T

= F V T

= T.

confidence rating #$&*: 3

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Given Solution:

`a**~p ^ ~q is false because ~q is false. One false is fatal to a conjunction.

~q is false so both parts of the disjunction [(~p^~q) U ~q] are false. Thus [(~p^~q) U ~q] is false.

The negation ~[(~p^~q) U ~q] of this statement is therefore true.**

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Self-critique (if necessary):

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Question: `q3.2.36 p: 15<8 q: 9 not > 5 r: 18 <= 18 evaluate -(p U -q) U -r

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Your solution:

- First I applied the given values to the variables.

= ~ [ (F) V ~ (T) ] V ~ (T)

Note that there are two ~’s in the statement; one for the given parenthetical quantity, and one inside attached to the T. This more or less reverts back to T anyway.

= ( T V T ) V F

= T V F

= T

CORRECTION: Didn’t read where 9 IS NOT greater than 4. Didn’t see the / across the >.

= ~ [ (F) V ~ (F) ] V ~ (T)

= (T V F) V F

= T V F

= T

confidence rating #$&*: 2

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Given Solution:

`a** p and q are both false statements, while r is a true statement.

It follows that p U ~q is true: since ~q is true the disjunction is true.

It therefore follows that ~(p U ~q) is false.

Since r is true, ~r is false.

Thus ~(p U ~q) U ~r is a disjunction of two false statements, ~(p U ~q) and ~r.

A disjunction of two false statements is false.

So the statement is false. **

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Self-critique (if necessary): How is the statement ~ ( p U ~ q) false? Perhaps I’m not seeing the forest for the trees, but doesn’t the *~* ( p U ~q ) insinuate, by the distributive property of multiplication, that the ~ applies to each term?

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Self-critique Rating:

@&

That's a good conjecture, but the ~ does not distribute.

It sort of distributes, as you'll see shortly when we get to deMorgan's Laws , but in the process the U becomes a ^ and a ^ becomes a U.

You'll understand this very well when we get to it. In the meantime you'll need to evaluate the expressions from the inside out.

*@

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Question: `q (formerly 3.2.42) This wasn't assigned, but you should be able to answer based on your responses to similar assigned questions. {}{}How many rows are there in a statement involving p,q,r,s,u,v,m,n? Note that rows go across the page. For example a statement involving just p and q will have four rows, one each for TT, TF, FT and FF. The headings (i.e., p, q and whatever other statements are necessary to evaluate the truth table) might also be considered a row, but for this problem do not consider the headings to be a row.

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Your solution:

- For every set of truths, 2, raised to the power of total terms, 8, to equal 256.

confidence rating #$&*:

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Given Solution:

`a** If you just have two statements p and q, then there are four possible truth values: TT, TF, FT and FF.

If you have three statements p, q and r then there are eight possible truth values: TTT, TTF, TFT, TFF, and FTT, FTF, FFT, FFF.

Note that the number of possible truth values doubles every time you add a statement.

The number of truth values for 2 statements is 4, which is 2^2.

For 3 statements this doubles to 8, which is 2^3.

Every added statement doubles the number, which adds a power to 2.

From this we see that the number of possible truth values for n statements is 2^n.

For the 8 statements listed for this problem, there are therefore 2^8 =256 possible truth values. **

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Self-critique (if necessary):

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Question: `q3.2.56 (fomerly 3.2.54) This was not assigned but based on your work on similar problems you should be able to construct the truth table for (-p ^ -q) U (~p U q). Give your truth table:

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Your solution:

P q (~p ^

T T

T F -

F T -

F F -

-

confidence rating #$&*: 1

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Given Solution:

`a** For column headings

p q ~p ~q ~p^~q ~p U q (~p^~q) U (~p Uq)

the first row would start off T T, for p and for q. Then F F for ~p and ~q. Then F for ~p ^ ~q, then T for ~p V q, then T for the final column.

So the first row would be

T T F F F T T.

The second row would be

T F F T F F F

The third row would be

F T T F F T T

and the fourth row would be

F F T T T T T **

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Self-critique (if necessary): Please provide a paraphrasal on how to complete this type of problem. My method was to plug in the values for p and q as (T, T) , (T, F) , (F, T) , (F, F) then provide the solution under the second column, labeled as (~p^~q) U (~p Uq).

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Self-critique Rating: 1

@&

When you evaluate

(~p^~q) U (~p Uq)

you'll want to have columns for both

(~p^~q)

and for

(~p Uq).

So you should have columns with these headings, preceding the column for (~p^~q) U (~p Uq) . To evaluate (~p^~q) U (~p Uq), then, you need only apply the rule for disjunction to the truth values in those two columns.

When you evaluate

(~p^~q)

you will want to refer to columns labeled ~p and ~q. If those columns are there, you need only apply the rule for conjuction in order to evaluate (~p^~q).

When you evaluate

(~p Uq)

you will want to be able to refer to columns for ~p and for q. You will probably already have the column for ~p listed, since it was used to evaluate (~p^~q). And the column for q would have been listed at the beginning.

With the columns headed

p, q, ~p, ~q, ~p^~q, ~p U q, (~p^~q) U (~p Uq)

as indicated in the given solution, then, each column can be evaluated by reference to two of the preceding columns.

You can also see this explained on the DVD.

*@

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Question: `q3.2.68 (formerly 3.2.66) This wasn't assigned but is similar to other assigned problems so you should be able to solve it: Negate the following statement using De Morgan's Law: ' F.C. tried to sell the wine but was unable to do so'.

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Your solution:

- F.C. didn’t try to sell the wine or wasn’t able to do so.

confidence rating #$&*: 1

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Given Solution:

`a** We use two ideas here. The first is that 'but' is interpreted as 'and'; and the second is that the negation of an 'and' statement is an 'or' statement. deMorgan's Laws say that the negation of p OR q is ~p AND ~q, while the negation of p AND q is ~p OR ~q.

The given statement ' F.C. tried to sell the book but was unable to do so' can be symbolized as 'p ^ q'. Its negation would be ~(p ^ q) = ~p U ~q. We translate this as 'F.C. didn't try to sell the book or he sold it', or something equivalent. **

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Self-critique (if necessary): Need an opinion on how to better understand these statements. Otherwise feel completely lost.

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Question: `q3.2.81 is the statement 3 + 1 = 6 xor 2 + 5 = 7 true or false?

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Your solution:

- Because this is an XOR case we see that we need only one true statement. Since one is true and the other is false, this is true.

confidence rating #$&*:

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Given Solution:

`a** For an XOR statement exactly one part has to be true. The statement is true because the second part is true and the first is false.

We need exactly one true statement; if both parts were true the XOR wouldn't be. **

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Self-critique (if necessary):

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Question: `q3.2.81 is the statement 3 + 1 = 6 xor 2 + 5 = 7 true or false?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

- Because this is an XOR case we see that we need only one true statement. Since one is true and the other is false, this is true.

confidence rating #$&*:

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Given Solution:

`a** For an XOR statement exactly one part has to be true. The statement is true because the second part is true and the first is false.

We need exactly one true statement; if both parts were true the XOR wouldn't be. **

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Self-critique (if necessary):

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Self-critique rating:

#*&!

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Question: `q3.2.81 is the statement 3 + 1 = 6 xor 2 + 5 = 7 true or false?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

- Because this is an XOR case we see that we need only one true statement. Since one is true and the other is false, this is true.

confidence rating #$&*:

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Given Solution:

`a** For an XOR statement exactly one part has to be true. The statement is true because the second part is true and the first is false.

We need exactly one true statement; if both parts were true the XOR wouldn't be. **

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Self-critique (if necessary):

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Self-critique rating:

#*&!#*&!

&#Good responses. See my notes and let me know if you have questions. &#