May 17th, 2018, 08:01 PM  #1 
Newbie Joined: May 2018 From: notimportant Posts: 9 Thanks: 0  Propositional Logic
Please help me with this thing. I'm so frustrated I can't understand propositional logic Demonstrate this: (p ∧ q) ↓ q ≡ ¬q PLEASE. 
May 17th, 2018, 08:09 PM  #2  
Senior Member Joined: Aug 2012 Posts: 2,193 Thanks: 645  Quote:
However with that interpretation of down arrow, your statement is not true. On the right you have $q \equiv \neg q$. That is tautologically false. On the left, $p \land q$ is true just in case both $p$ and $q$ are true. So if $p$ is true and $q$ is true, then the left hand side of the down arrow is true, and the right hand side is false. So the down arrow expression is false. Unless you are interpreting the down arrow with a different meaning, this is how I see it.  
May 17th, 2018, 08:34 PM  #3  
Newbie Joined: May 2018 From: notimportant Posts: 9 Thanks: 0  Quote:
Thanks., maybe you can solve that with the laws of logic.?  
May 17th, 2018, 08:58 PM  #4 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,923 Thanks: 1122 Math Focus: Elementary mathematics and beyond 
I'm looking at [(p ∧ q) ↓ q] $\implies$ ¬q, aren't I? That's true, isn't it?

May 17th, 2018, 09:19 PM  #5 
Newbie Joined: May 2018 From: notimportant Posts: 9 Thanks: 0  
May 17th, 2018, 09:27 PM  #6 
Senior Member Joined: Sep 2015 From: USA Posts: 2,369 Thanks: 1273 
$\begin{align*} &(p \wedge q) \downarrow q \equiv \\ \\ &\neg((p\wedge q) \vee q) \equiv \\ \\ &\neg q \end{align*}$ The last from the fact that $(p \wedge q) \subseteq q$ 
May 17th, 2018, 09:32 PM  #7 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,923 Thanks: 1122 Math Focus: Elementary mathematics and beyond 
The three lines mean "logically equivalent to". That is, the statement to the left of the three lines is logically equivalent to the statement to the right of the three lines. If your problem was interpreted that way, it would appear valid.

May 17th, 2018, 09:34 PM  #8 
Newbie Joined: May 2018 From: notimportant Posts: 9 Thanks: 0  
May 17th, 2018, 10:04 PM  #9  
Newbie Joined: May 2018 From: notimportant Posts: 9 Thanks: 0  Quote:
Thanks.  
May 18th, 2018, 12:50 AM  #10  
Senior Member Joined: Aug 2012 Posts: 2,193 Thanks: 645  Quote:
Wiki shows the order of precedence of the logical operators. Downarrow is not listed. https://en.wikipedia.org/wiki/Logica..._of_precedence Logical equivalence has the lowest precedence of all operators, so if we extend this principle to nor then the proposed solution is correct, that is we evaluate downarrow before equivalence. I Googled around and could not find any definitive resolution to this question. I found one Stackoverflow thread that notes that if an operator's precedence is not defined, it should be processed left to right and the proposed solution is correct. Still, this is all pretty murky. It depends on the order of precedence of an operator so obscure that its order of precedence is not explicitly defined anywhere on the Internet. One has to apply the principle that logical equivalence is always lower than any other operator anyone can think of. https://stackoverflow.com/questions/...nandnorxnor Last edited by Maschke; May 18th, 2018 at 01:12 AM.  

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