Assignment 13Query13

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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:

p and q would be true

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): OK

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Self-critique Rating: 3

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

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Your solution:

Both parts are false but where the entire thing is enclosed in a bracket with a ~, it makes the entire thing true

confidence rating #$&*:

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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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Self-critique Rating:

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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:

After working it out, I could find that ~r is false. However, I found that ~(p U ~q) was true.

@&

To evaluate ~(p U ~q) you have to first evaluate p U ~q.

To evaluate p U ~q you have to know the truth values of both p and ~q.

It's not hard to see that p is false and q is false.

So ~q is true.

Therefore p U ~q is the disjunction of a false and a true statement.

The rule for disjunction is that unless both statements are false, it's true.

Since p is false and ~q is true the disjunction is of a false and a true statement.

So the disjunction p U ~q is true.

Of course we're trying to evaluate

~(p U ~q)

Since p U ~q is true, it follows that ~(p U ~q) is false.

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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): Got part right. I’m having a hard time understanding.

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Self-critique Rating: 2

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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:

2^8 = 256 rows

confidence rating #$&*: 3

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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): OK

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Self-critique Rating: 3

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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:

F F F F F F T

F F T T F T F

T T F T T T T

T F T T T T F

@&

I can't interpret your table because you haven't specified what the headings are.

However the first two headings should be your values of p and q.

So when you start out your table should look like this:

p q

T T

T F

F T

F F

Since your expressions involve both ~p and ~q, you need to evaluate these expressions next. Your table would thus be expanded to something like the following:

p q ~p ~q

T T F

T F F

F T T

F F T

I've filled in the values of ~p, but I haven't filled in the values for ~q.

Can you fill those in?

Having done so you would then want to evaluate, say, ~p ^ q. To do so you would add another column to the table, which would look like this:

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

T T F

T F F

F T T

F F T

For each row, the value of ~p ^ ~q can quickly be determined by looking back and the values of ~p and ~q in that row, and using the rule for conjunction.

Can you fill in this table?

You would then need to add a column for (~p U q) and evaluate it based on the values of ~p and q.

Having filled in that column, you would add a final column for (-p ^ -q) U (~p U q)

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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): I don’t know what I’m doing wrong. I’m looking from examples in the book and I still can’t seem to understand.

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Self-critique Rating: 0

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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. tried to sell the wine or he was able to sell the wine.’

confidence rating #$&*: 2

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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): OK

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Self-critique Rating: 3

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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:

True; The first statement is false but the second is true.

confidence rating #$&*: 3

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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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end document

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See my note on the truth table. I recommend that if you're still not sure of what you're doing, you insert your best attempt at answers to my questions into a copy of this document, as you see it posted, and submit it.

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