Query 13

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course Mth 151

013. `query 13

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

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Your solution:

If I understand this correctly, if ~(p^q) is false then the truth is p U q or you can do ~p ^ ~q.

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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):

Did I do this right????? It looks like I did, but not sure.

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Self-critique Rating:

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

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Your solution:

~ [(T ^ F) U F]

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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):

I am really lost here, by reading the solution; I think I understand what needs to be done. I really need to come by and talk with you so I can get a better understanding.

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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:

If I go by what the book says the following should be correct:

~ (p U ~ q) U ~ r = False. Because the p is true, the ~q is false because of the conjunction you it will make the statement false. The ~ r is false. Two false makes the statement fasle.

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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):

Yea I got it right, not sure if I explained it exactly like it was suppose to be though.

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Self-critique Rating:

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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:

I believe if reading the chapter correctly the total rows that are needed for this true table would be 256. Because since we have 8 letters, you would have 2 to the 8th power giving you 256.

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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):

Yeah me!!!! I might be starting to get this chapter after all.

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Self-critique Rating:

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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:

This one wasn’t fun at all, and like most of this chapter I am not really grasping this concept. I do understand that fact that every p or q you will have a TT, TF, FT, FF ( but for the rest of the chart regarding the U or ^, I am not quite sure what to do when it comes to them.

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

T T F T F F F T T

T F F F T F F F F

F T T T T F T F F

F F T F F T T T T

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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):

I know I am repeating myself, but I think I got it right, just not sure.

@& F T F in your first row, under ~p ^ ~q, is incorrect. You are right that ~p and ~q are both F, but F ^ F is false. So that part should read 'F F F'.

In the third row TTT is not correct. ~q is false in that line so the third value should be F. This would then make ~p ^ q false. So instead of TTT you would have T F F.

In the fourth row TFF is not correct. ~q is T in that row, as is ~p, which would make ~p ^ ~q true. So where you ave TFF, you should have TTT.

You have some other errors, but first be sure you understand these. *@

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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:

Once again I am so not getting this. This chapter has been a very confusing one. But I want to say the answer would ~p ^ q or ‘F.C didn’t try to sell the wine but was able to do so’

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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'

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Self-critique (if necessary):

You will have to tell me if I got this one or not. I think I got the concept of the problem, but didn’t get it all right. I am so sorry.

@& Your statement is right except you used ^ where you should have used U

Logically, 'but' is the same as 'and'. So the original statement is an 'and' statement.

When you negate an 'and' statement you get an 'or' statement.*@

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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:

3 + 1 = 6 is False and 2 + 5 = 7 is true

@& That's right, and you have correctly evaluated the two statements.

But you have one more step, which is to evaluate the XOR.*@

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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. **

@& See my notes. I'll respond with some additional questions by sometime tomorrow, and we'll see what we can do to make this clearer for you.

You are aware of the rules, which is a very good first step, but you need some practice to learn to apply them effectively.*@