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course MTH 151
9:20pm, 3/4/14
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.
014. Truth Tables
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Question: `q001. There are 10 questions in this set.
If each of the propositions p and q can be either true or false, what combinations of truth values are possible for the two propositions (e.g., one possibility is that p is false and q is true; list the other possibilities)?
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
P is false and q is true; p is true and q is false; both could be true; both could be false
confidence rating #$&*: 3
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Given Solution:
It is possible that p is true and q is true.
Another possibility is that p is true and q is false.
A third possibility is that p is false and q is true.
A fourth possibility is that p is false and q is false.
These possibilities can be listed as TT, TF, FT and FF, where it is understood that the first truth value is for p and the second for q.
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Self-critique (if necessary):
ok
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Self-critique Rating: 3
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Question: `q002. For each of the for possibilities TT, TF, FT and FF, what is the truth value of the compound statement p ^ q ?
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
If both p and q were true, p ^ q would be right. If both were false, p ^ q would be false.
confidence rating #$&*: 2
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Given Solution:
p ^ q means 'p and q', which is only true if both p and q are true.
In the case TT, p is true and q is true so p ^ q is true.
In the case TF, p is true and q is false so p ^ q is false.
In the case FT, p is false and q is true so p ^ q is false.
In the case FF, p is false and q is false so p ^ q is false.
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Self-critique (if necessary):
I found this question to be a little confusing.
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&#Your response did not agree with the given solution in all details, and you should therefore have addressed the discrepancy with a full self-critique, detailing the discrepancy and demonstrating exactly what you do and do not understand about the parts of the given solution on which your solution didn't agree, and if necessary asking specific questions (to which I will respond).
&#
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Self-critique Rating: 3
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Question: `q003. Write the results of the preceding problem in the form of a truth table.
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
TTT; FFF; FTF; TFT;
confidence rating #$&*: 2
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Given Solution:
The truth table must have headings for p, q and p ^ q. It must include a line for each of the possible combinations of truth values for p and q. The table is as follows:
p q p ^ q
T T T
T F F
F T F
F F F.
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Self-critique (if necessary):
Finding this question confusing as well.
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&#This also requires a self-critique.
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Self-critique Rating: 3
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Question: `q004. For each of the possible combinations TT, TF, FT, FF, what is the truth value of the proposition p ^ ~q?
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
If p and q are both true, both ~p and ~q would be false; if p and q are both false, then ~p and ~q would both be true. P ^ ~q, to me, seems to read that p is true and q is false.
confidence rating #$&*: 2
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Given Solution:
For TT we have p true, q true so ~q is false and p ^ ~q is false.
For TF we have p true, q false so ~q is true and p ^ ~q is true.
For FT we have p false, q true so ~q is false and p ^ ~q is false.
For FF we have p false, q false so ~q is true and p ^ ~q is false.
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Self-critique (if necessary):
I have a hard time understanding how to write the truth statement when one is false and one is true, and finding which is false and which is true. For example, TF is p true, q false so ~q is true and p ^ ~q is true <- how do you know that p ^ ~q is true, and how is this determined?
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Self-critique Rating: 3
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You evaluate the statement for each possible combination of truth values.
The truth values apply, in this case, to p and q, in that order.
So TF means that p is true and q is false.
In this case, since q is false ~q (which means 'the negation of q' or just 'not q') is true.
The '^' symbol is used here to stand for conjunction (roughly interpreted as 'and'). So p ^ ~q is a conjunction of p and ~q. For the TF can, both of these statements are true, so their conjunction is true.
Thus, in te TF case, p is true, q is false so ~q is true, and p ^ ~q is therefore true.
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Question: `q005. Give the results of the preceding question in the form of a truth table.
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
TTFF, FFTT…
confidence rating #$&*: 1
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Given Solution:
The truth table will have to have headings for p, q, ~q and p ^ ~q. We therefore have the following:
p q ~q p^~q
T T F F
T F T T
F T F F
F F T F
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Self-critique (if necessary):
I’m not sure why I’m having so much trouble understanding
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Self-critique Rating: 2
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You start the truth table by listing the possible truth values of p and q:
p q
T T
T F
F T
F F
Then in this case you proceed to list the corresponding values of ~q. For each line the value of ~q is the opposite of the value of q, so at this point the table would read
p q ~q
T T F
T F T
F T F
F F T
Finallyl you add a column for p ^ ~q, which gives you the truth values in the given solution.
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@&
You start the truth table by listing the possible truth values of p and q:
p q
T T
T F
F T
F F
Then in this case you proceed to list the corresponding values of ~q. For each line the value of ~q is the opposite of the value of q, so at this point the table would read
p q ~q
T T F
T F T
F T F
F F T
Finallyl you add a column for p ^ ~q, which gives you the truth values in the given solution.
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Question: `q006. Give the truth table for the proposition p U q, where U stands for disjunction.
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
TTT; FTF; TTF;
confidence rating #$&*: 2
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Given Solution:
p U q means 'p or q' and is true whenever at least one of the statements p, q is true. Therefore p U q is true in the cases TT, TF, FT, all of which have at least one 'true', and false in the case FF. The truth table therefore reads
p q p U q
T T T
T F T
F T T
F F F
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Self-critique (if necessary):
I didn’t realize that FFF would be included, because I thought that there had to be at least one true statement.
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You start by listing all the possible combinations of truth values of p and q (see also my preceding note).
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Self-critique Rating: 3
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Question: `q007. Reason out the truth values of the proposition ~(pU~q).
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
My guess would be that if p is true, that ~p would be false, and if ~q is true, then q would be false.
confidence rating #$&*: 1
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Given Solution:
In the case TT p is true and q is true, so ~q is false. Thus p U ~q is true, since p is true. So ~(p U ~q) is false.
In the case TF p is true and q is false, so ~q is true. Thus p U ~q is true, since p is true (as is q). So ~(p U ~q) is false.
In the case FT p is false and q is true, so ~q is false. Thus p U ~q is false, since neither p nor ~q is true. So ~(p U ~q) is true.
In the case FF p is false and q is false, so ~q is true. Thus p U ~q is true, since ~q is true. So ~(p U ~q) is false.
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Self-critique (if necessary):
I’m just having a lot of trouble understanding how to write the statements when there is a ~ inside the parenthesis, as well as outside the parenthesis. I wasn’t even sure how to begin to write an answer.
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~ changes the truth value of whatever it applies to. If it is in front of q, is changes the truth value of q. If it is in front of (p U ~q), then it changes the truth value of ; (p U ~q)
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Self-critique Rating: 3
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You need to start by listing the values of p and q, then step by step you construct the table for the statement you are analyzing.
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YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
TTFTT, TTFFF…
confidence rating #$&*: 1
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Given Solution:
We need headings for p, q, ~q, p U ~q and ~(p U ~q). Our truth table therefore read as follows:
p q ~q pU~q ~(pU~q)
T T F T F
T F T T F
F T F F T
F F T T F
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Self-critique (if necessary):
Still having a hard time being able to understand the use of ‘~’, and how to interpret it and see how it is true/false, when the others are true/false.
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Self-critique Rating: 3
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Question: `q009. Construct a truth table for the statement (p ^ ~q).
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
If I had to guess, I would say TTF, TFT, and TFF.
@&
As illustrated in the solutions, the truth table starts with the headings of the columns. The first two columns will have headings p and q.
The table is then constructed accordingly, line by line.
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confidence rating #$&*:
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Question: `q010. Construct a truth table for the statement q U (p ^ ~q).
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
If guessing, TTTFF, FFFTT…
confidence rating #$&*:
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Self-critique Rating:
@&
You aren't listing truth tables in the specified form, and this might have a lot to do with your difficulties in understanding what's going on here.
I expect that the text will help you sort this out, but I've inserted a number of notes to hopefully clarify some essential points and help you in the process.
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