Question 147603
Can someone help me construct this truth table? P->(P ->(Q ^ P)). Now, does this mean that first, you work the Q^P? or do you work the p-> which is inside the ( ) first. Then do you take the result of whichever is used and connect it with p-> which is oustide the ( ). Do that make sense. I'm confused too.
<pre><font size = 4 color = "indigo"><b>
Start with this:

<u>P|Q| P->(P ->(Q ^ P))</u>
T|T|   
T|F|    
F|T|  
F|F| 

Now transfer TTFF under every P, and TFTF under every Q

<u>P|Q| P->(P ->(Q ^ P))</u>
T|T| T   T    T   T  
T|F| T   T    F   T   
F|T| F   F    T   F  
F|F| F   F    F   F

Now take care of the ^, because that's the innermost
operation.  If ^ is between two T's it's T, otherwise it's F
So we use this rule to fill in under the ^, but everytime
we do we scratch through the letters on each side of it,
like this:

<u>P|Q| P->(P ->(Q ^ P))</u>
T|T| T   T    <s>T</s> T <s>T</s>  
T|F| T   T    <s>F</s> F <s>T</s>   
F|T| F   F    <s>T</s> F <s>F</s>  
F|F| F   F    <s>F</s> F <s>F</s>

Now I'll just erase them, but you can just leave them
scratched:

<u>P|Q| P->(P ->(Q ^ P))</u>
T|T| T   T      T    
T|F| T   T      F     
F|T| F   F      F    
F|F| F   F      F

Now take care of the second ->, because that's now the innermost
operation.  If -> has a T on the left and an F on the right,
then it's F, otherwise it's T. So we use this rule to fill 
in under the ->, but as before, every time we do we scratch 
through the letters on each side of it, like this:  

<u>P|Q| P->(P ->(Q ^ P))</u>
T|T| T   <s>T</s>  T   <s>T</s>    
T|F| T   <s>T</s>  F   <s>F</s>     
F|T| F   <s>F</s>  T   <s>F</s>    
F|F| F   <s>F</s>  T   <s>F</s>

Again I'll erase the ones scratched, but you can just leave them
scratched:

<u>P|Q| P->(P ->(Q ^ P))</u>
T|T| T      T       
T|F| T      F        
F|T| F      T       
F|F| F      T   

Finally we take care of the first ->, because that's now 
the innermost operation.  We use the same rule again to fill 
in under the first ->,

<u>P|Q| P->(P ->(Q ^ P))</u>
T|T| <s>T</s> T    <s>T</s>       
T|F| <s>T</s> F    <s>F</s>        
F|T| <s>F</s> T    <s>T</s>       
F|F| <s>F</s> T    <s>T</s>   

And erasing those marked thru:

<u>P|Q| P->(P ->(Q ^ P))</u>
T|T|   T           
T|F|   F            
F|T|   T           
F|F|   T     

Edwin</pre>