course Mth 151
f 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.
017. Evaluating Arguments
&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
Self-critique (if necessary):
Self-critique Rating:
Question: **** `q001. There are 9 questions in this set.
Explain why [ (p -> q) ^ (q -> r) ^ p] -> r must be true for every set of truth values for which r is true.
*********************************************
Your solution:
because it is a tautology?
Confidence Assessment:
*********************************************
Given Solution:
[ (p -> q) ^ (q -> r) ^ p] -> r must be true if r is true, since the statement can only be false if [ (p -> q) ^ (q -> r) ^ p] is true while r is false. Therefore the truth values TTT, TFT, FTT, FFT (i.e., all the truth values that have r true) all make the statement true.
&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
Self-critique (if necessary):
I still don? quite grasp this. Is this the same thing as the table?
You could make a table, which would be useful in understanding the above explanation.
You should break the given explanation up into phrases and tell me what you do and do not understand about each. For example:
[ (p -> q) ^ (q -> r) ^ p] -> r must be true if r is true (Do you understand what this is saying? Do you understand why it must be so? Exactly what do you understand and what do you not undersatnd about this statement?)
the statement can only be false if [ (p -> q) ^ (q -> r) ^ p] is true while r is false. (Do you understand what this is saying? Do you understand why it must be so? Exactly what do you understand and what do you not undersatnd about this statement?)
TTT, TFT, FTT, FFT are all the truth values that have r true (Do you understand what this is saying? Do you understand why it must be so? Exactly what do you understand and what do you not undersatnd about this statement?)
the truth values TTT, TFT, FTT, FFT (i.e., all the truth values that have r true) all make the statement true. (Do you understand what this is saying? Do you understand why it must be so? Exactly what do you understand and what do you not undersatnd about this statement?)
[ (p -> q) ^ (q -> r) ^ p] -> r must be true if r is true, since the statement can only be false if [ (p -> q) ^ (q -> r) ^ p] is true while r is false. (Do you understand what this is saying? Do you understand why it must be so? Exactly what do you understand and what do you not undersatnd about this statement?)
Now putting it all together: [ (p -> q) ^ (q -> r) ^ p] -> r must be true if r is true, since the statement can only be false if [ (p -> q) ^ (q -> r) ^ p] is true while r is false. Therefore the truth values TTT, TFT, FTT, FFT (i.e., all the truth values that have r true) all make the statement true. (Do you understand what this is saying? Do you understand why it must be so? Exactly what do you understand and what do you not undersatnd about this statement?)
Self-critique Rating:
Question: **** `q002. At this point we know that the truth values TTT, TFT, FTT, FFT all make the argument [ (p -> q) ^ (q -> r) ^ p] -> r true. What about the truth values TTF?
*********************************************
Your solution:
I think it is the way it is set up that confuses me. If I see it in a table it would be clearer.
Confidence Assessment:
*********************************************
Given Solution:
It would be possible to evaluate every one of the statements p -> q, q -> r, etc. for their truth values, given truth values TTF. However we can shortcut the process.
We see that [ (p -> q) ^ (q -> r) ^ p] is a compound statement with conjunction ^. This means that [ (p -> q) ^ (q -> r) ^ p] will be false if any one of the three compound statements p -> q, q -> r, p is false.
For TTF we see that one of these statements is false, so that [ (p -> q) ^ (q -> r) ^ p] is false. This therefore makes the statement [ (p -> q) ^ (q -> r) ^ p] -> r true.
&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
Self-critique (if necessary):
This is just what the book said when I was trying to answer it. It said if at least one premise in a conjunction of several premises is false then the entire conjunction is false.
Self-critique Rating:
Question: **** `q003. The preceding statement said that for the TTF case [ (p -> q) ^ (q -> r) ^ p] was false but did not provide an explanation of this statement. Which of the statements is false for the truth values TTF, and what does this tell us about the truth of the statement [ (p -> q) ^ (q -> r) ^ p] -> r?
*********************************************
Your solution:
p q r p->q q->r [(p->q)^(q->r)^p] ->r
T T T T T T T
T T F T F T F
T F T F T T T
T F F F T T F
F T T T T T T
F T F T F T F
F F T T T T T
F F F T T T F
because it is a tautology.
Confidence Assessment:
*********************************************
Given Solution:
p and q are both true, so p -> q and p are true. The only candidate for a false statement among the three statements is q -> r.
So we evaluate q -> r for truth values TTF. Since q is T and r is F, we see that q -> r must be F.
This makes [ (p -> q) ^ (q -> r) ^ p] false. Therefore [ (p -> q) ^ (q -> r) ^ p] -> r must be true, since it can only be false and if [ (p -> q) ^ (q -> r) ^ p] is true.
&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
Self-critique (if necessary):
Explain to me about finding truth in these sets such as TTF. I can? find it in the book nor did the lady on the video say anything about them. If it is in the book can you tell me what page? When I figure it out I will finish this assignment and resubmit it.
TTF stands for the truth values of p, q and r. TTF means the p is true, q is true and r is false.
In your truth table this corresponds to the second line, which should read:
p q r p->q q->r [(p->q)^(q->r)^p] [(p->q)^(q->r)^p]->r
T T F T F F T
Note that [(p->q)^(q->r)^p] is false for this line, because this expression is a conjunction and at least one of the statement s in the conjunction (namely q -> r) is false. This makes [(p->q)^(q->r)^p] -> r true, since a false antecedent makes the conditional true.