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princesspixie
Group Title
show that the statement (~p V q) ^ (p^~q) is a logical contradiction
 one year ago
 one year ago
princesspixie Group Title
show that the statement (~p V q) ^ (p^~q) is a logical contradiction
 one year ago
 one year ago

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KingGeorge Group TitleBest ResponseYou've already chosen the best response.2
Have you been taught how to do truth tables?
 one year ago

KingGeorge Group TitleBest ResponseYou've already chosen the best response.2
Well, they make the problem easier, but you can still do it without them. What we need to do, is break the problem up into 4 different cases. Case 1: p=True, q=True. Case 2: p=True, q=False. Case 3: p=False, q=True. Case 4: p=False, q=False. I'll work through the first case.
 one year ago

princesspixie Group TitleBest ResponseYou've already chosen the best response.0
ive seen them in my text book but i am not sure how to do one
 one year ago

KingGeorge Group TitleBest ResponseYou've already chosen the best response.2
Suppose p=True, and q=True. Then (~p V q) is the same as (False OR True), which is True, since q is True. Also, (p ^ ~q) is the same as (True AND False), which is False, since q and p have different truth values (one is true, the other is false). Thus, we end with True AND False, which by the same reasoning, is False. To do the rest, you need to go through each case, and show that you always end with False. Can you try to do Case 2 by yourself?
 one year ago

UnkleRhaukus Group TitleBest ResponseYou've already chosen the best response.1
if you were gong to use a truth table set it up like this: \[\begin{array}{ccccc}\hline p&\neg p&q&\neg q&\neg p\lor q&p\land \neg q&(\neg p\lor q)\land(p\land \neg q)\\\hline T&&T\\T&&F \\F&&T\\T&&F\\\hline\end{array}\]
 one year ago

princesspixie Group TitleBest ResponseYou've already chosen the best response.0
@KingGeorge i am not even sure where to start...
 one year ago

KingGeorge Group TitleBest ResponseYou've already chosen the best response.2
Alright. So start with Case 2: p is True, and q is False. Now let's take it one step at a time. What is ~p?
 one year ago

princesspixie Group TitleBest ResponseYou've already chosen the best response.0
false
 one year ago

KingGeorge Group TitleBest ResponseYou've already chosen the best response.2
right. Now, ~p is False, and q is true. What is ~p OR q?
 one year ago

princesspixie Group TitleBest ResponseYou've already chosen the best response.0
false
 one year ago

KingGeorge Group TitleBest ResponseYou've already chosen the best response.2
Not so fast. Remember that q is True, and since you only need one thing to be true in an OR statement, ~p OR q is actually True.
 one year ago

KingGeorge Group TitleBest ResponseYou've already chosen the best response.2
Let's go to the next step. What is ~q?
 one year ago

princesspixie Group TitleBest ResponseYou've already chosen the best response.0
I dont get it..
 one year ago

princesspixie Group TitleBest ResponseYou've already chosen the best response.0
true
 one year ago

KingGeorge Group TitleBest ResponseYou've already chosen the best response.2
Alright, let's take a step back, and look at ~p OR q again. For (~p OR q) to be true, either ~p is True, or q is True. Since q is True, (~p OR q) is True. Does this make more sense?
 one year ago

princesspixie Group TitleBest ResponseYou've already chosen the best response.0
i guess
 one year ago

KingGeorge Group TitleBest ResponseYou've already chosen the best response.2
Alright, stop me if you start to not understand anything. Now, q is True. So what is ~q?
 one year ago

princesspixie Group TitleBest ResponseYou've already chosen the best response.0
false
 one year ago

KingGeorge Group TitleBest ResponseYou've already chosen the best response.2
Precisely. Now, recall that p is True, and ~q is False. Can you tell me what (p AND ~q) is?
 one year ago

princesspixie Group TitleBest ResponseYou've already chosen the best response.0
false
 one year ago

KingGeorge Group TitleBest ResponseYou've already chosen the best response.2
oops, I screwed up a bit.
 one year ago

KingGeorge Group TitleBest ResponseYou've already chosen the best response.2
Since we're assuming p is true, and q is False, ~q is True. So, (p AND ~q) is actually...?
 one year ago

princesspixie Group TitleBest ResponseYou've already chosen the best response.0
trur
 one year ago

KingGeorge Group TitleBest ResponseYou've already chosen the best response.2
Oh wow. I screwed up earlier as well. You were correct in saying that (~p OR q) is False. Anyways, then we only have (~p OR q) AND (p AND ~q) left. This simplifies to (False AND True). What is the truth value of this?
 one year ago

princesspixie Group TitleBest ResponseYou've already chosen the best response.0
false true
 one year ago

KingGeorge Group TitleBest ResponseYou've already chosen the best response.2
Well, for (False AND True) to be True, either both need to be false, or both need to be true. Since neither is the case, (False AND True) is False. Now, I'm going to fill in the truth table that UnkleRhaukus kindly provided.
 one year ago

KingGeorge Group TitleBest ResponseYou've already chosen the best response.2
\[\begin{array}{ccccc}\hline p&\neg p&q&\neg q&\neg p\lor q&p\land \neg q&(\neg p\lor q)\land(p\land \neg q)\\\hline T&F&T&F&T&F&F\\T&F&F&T&F&T&F \\F&T&T&F\\T&T&F&T\\\hline\end{array}\] The first two lines correspond to Case 1 and Case 2 respectively. Now we want to do Case 3. So p is False, and q is True. Can you tell me what (~p OR q) is?
 one year ago

princesspixie Group TitleBest ResponseYou've already chosen the best response.0
true
 one year ago

KingGeorge Group TitleBest ResponseYou've already chosen the best response.2
right. what about (p AND ~q)?
 one year ago

princesspixie Group TitleBest ResponseYou've already chosen the best response.0
false since q is true
 one year ago

KingGeorge Group TitleBest ResponseYou've already chosen the best response.2
Bingo. Then, what about (~p OR q) AND (p AND ~q)?
 one year ago

princesspixie Group TitleBest ResponseYou've already chosen the best response.0
true
 one year ago

UnkleRhaukus Group TitleBest ResponseYou've already chosen the best response.1
whops i set up the table with a mistake \[\begin{array}{ccccc}\hline p&\neg p&q&\neg q&\neg p\lor q&p\land \neg q&(\neg p\lor q)\land(p\land \neg q)\\\hline T&F&T&F&T&F&F\\T&F&F&T&F&T&F \\F&T&T&F\\\color{red}F&T&F&T\\\hline\end{array}\]
 one year ago

KingGeorge Group TitleBest ResponseYou've already chosen the best response.2
Not quite. Remember that (~p OR q) is True, but (q AND ~q) is False. Since they're different truth values, (~p OR q) AND (p AND ~q) must be False.
 one year ago

KingGeorge Group TitleBest ResponseYou've already chosen the best response.2
Good catch^^
 one year ago

KingGeorge Group TitleBest ResponseYou've already chosen the best response.2
Now we just have the final case, where p is False, and q is False. Can you work through that, and tell me what you think (~p OR q) is, and what (p AND ~q) is?
 one year ago

princesspixie Group TitleBest ResponseYou've already chosen the best response.0
false
 one year ago

princesspixie Group TitleBest ResponseYou've already chosen the best response.0
@KingGeorge
 one year ago

KingGeorge Group TitleBest ResponseYou've already chosen the best response.2
What exactly are you saying is false?
 one year ago

princesspixie Group TitleBest ResponseYou've already chosen the best response.0
~p or q
 one year ago

princesspixie Group TitleBest ResponseYou've already chosen the best response.0
actually i mean that one is true since p is false
 one year ago

KingGeorge Group TitleBest ResponseYou've already chosen the best response.2
Right. what about (p AND ~q)?
 one year ago

princesspixie Group TitleBest ResponseYou've already chosen the best response.0
true
 one year ago

KingGeorge Group TitleBest ResponseYou've already chosen the best response.2
Remember that p is False, and ~q is True, so (p AND ~q) would actually be False.
 one year ago

KingGeorge Group TitleBest ResponseYou've already chosen the best response.2
Now here's the very last step. What is (~p OR q) AND (p AND ~q)?
 one year ago

princesspixie Group TitleBest ResponseYou've already chosen the best response.0
false and false?
 one year ago

KingGeorge Group TitleBest ResponseYou've already chosen the best response.2
Well the statement as a whole is False. Now, if we look at the truth table, we get a table that looks like this. \[\begin{array}{ccccc}\hline p&\neg p&q&\neg q&\neg p\lor q&p\land \neg q&(\neg p\lor q)\land(p\land \neg q)\\\hline T&F&T&F&T&F&F\\T&F&F&T&F&T&F \\F&T&T&F&T&F&F\\F&T&F&T&T&F&F\\\hline\end{array}\]Since the entire last column is filled with F, we've shown that the original statement is a logical contradiction.
 one year ago

KingGeorge Group TitleBest ResponseYou've already chosen the best response.2
If you want to know some more about truth tables, and how to construct them, feel free to ask.
 one year ago

princesspixie Group TitleBest ResponseYou've already chosen the best response.0
ok great so that truth table is the whole answer?
 one year ago

princesspixie Group TitleBest ResponseYou've already chosen the best response.0
thanks so much! i have other questions but there not about truth tables
 one year ago

KingGeorge Group TitleBest ResponseYou've already chosen the best response.2
The truth table is a great way to show that you have a logical contradiction, and one of the easiest ways to see it directly.
 one year ago

princesspixie Group TitleBest ResponseYou've already chosen the best response.0
dw:1355892045358:dw
 one year ago

princesspixie Group TitleBest ResponseYou've already chosen the best response.0
The number of rows in the truth table for the compound statement
 one year ago

princesspixie Group TitleBest ResponseYou've already chosen the best response.0
would the answer be 8?
 one year ago

KingGeorge Group TitleBest ResponseYou've already chosen the best response.2
First off,how many variables do you have?
 one year ago

KingGeorge Group TitleBest ResponseYou've already chosen the best response.2
I mean like p and q are variable. So in this context, it's a variable if it can be true or false.
 one year ago

KingGeorge Group TitleBest ResponseYou've already chosen the best response.2
How many of those are there?
 one year ago

KingGeorge Group TitleBest ResponseYou've already chosen the best response.2
Right. Then for these truth problems, each variable can be either T or F, and each row in the truth table corresponds to a different combination of these possibilities. Since each variable has 2 possibilities, and there are 4 variables, there are \(2\cdot2\cdot2\cdot2=16\) rows in the truth table. In general, for a truth statement with \(k\) variables, there would be \[2^k\] rows in the truth table.
 one year ago

princesspixie Group TitleBest ResponseYou've already chosen the best response.0
so it would be 16?
 one year ago

princesspixie Group TitleBest ResponseYou've already chosen the best response.0
great do you now anything about contrapositive statments?
 one year ago

KingGeorge Group TitleBest ResponseYou've already chosen the best response.2
Sure. If you have more questions though, please put it in a new question.
 one year ago

princesspixie Group TitleBest ResponseYou've already chosen the best response.0
ok thanks!!
 one year ago
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