How Can Logic Laws Remove Implications in Expressions?

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Discussion Overview

The discussion revolves around the application of logic laws to remove implications in propositional expressions. Participants are exploring how to manipulate logical statements without using truth tables, focusing on the laws of propositional logic and their application in a university-level assignment.

Discussion Character

  • Homework-related, Technical explanation, Exploratory

Main Points Raised

  • One participant expresses confusion about removing implications from a logical expression and seeks clarification on how to apply the laws of logic.
  • Another participant suggests substituting implications with disjunctions as a first step in the problem-solving process.
  • A participant provides a detailed step-by-step manipulation of a logical expression, demonstrating the use of various laws of logic, including De Morgan's laws and distributive properties.
  • There is a request for clarification on the laws of propositional logic that can be utilized in the assignment.
  • One participant mentions feeling "completely in the dark" about the concept and indicates that the provided resources do not adequately explain the topic.
  • A later post presents a structured proof of a logical equivalence, showcasing the application of implication and other logical transformations.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the best approach to removing implications, as there are varying levels of understanding and different methods proposed. Some participants express confusion, while others provide detailed explanations and manipulations.

Contextual Notes

Participants indicate limitations in their understanding of the topic, with some expressing that their textbooks and lectures do not clarify how to apply the laws of logic effectively. There is also mention of the assignment's constraints, specifically the prohibition of using truth tables.

Who May Find This Useful

This discussion may be useful for students studying propositional logic, particularly those encountering challenges in applying logical laws to expressions in a homework context.

werebilby
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Hello Everyone,

This is my first post, so please me kind :D

I have an assignment and have been stuck on this thing for 2 days. I have no problems working this out with a truth table but we are not allowed to use this. So this it the expression.

View attachment 5619

I understand that I have to remove the implication but this doesn't really make sense to me and I have been using Youtube etc to try and get more info but just doesn't make sense as to how to apply the Laws of Logic. Help please?!

Regards
Werebilby
 

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    Q1(iv)2.PNG
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werebilby said:
Hello Everyone,

This is my first post, so please me kind :D

I have an assignment and have been stuck on this thing for 2 days. I have no problems working this out with a truth table but we are not allowed to use this. So this it the expression.
I understand that I have to remove the implication but this doesn't really make sense to me and I have been using Youtube etc to try and get more info but just doesn't make sense as to how to apply the Laws of Logic. Help please?!

Regards
Werebilby

Which are the laws of propositional logic that you can use
 
solakis said:
Which are the laws of propositional logic that you can use

Ok so the laws I can use are :-

View attachment 5620
 

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werebilby said:
Ok so the laws I can use are :-

O.k

Note some of the laws you have mentioned can be proved using the others.

Coming now to your problem the 1st step is to substitute the "=>" with "v"

Can you do that??
 
solakis said:
O.k

Note some of the laws you have mentioned can be proved using the others.

Coming now to your problem the 1st step is to substitute the "=>" with "v"

Can you do that??

Ok let me try that again. So this is what I have worked out so far.

View attachment 5623
 

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werebilby said:
Ok let me try that again. So this is what I have worked out so far.

http://mathhelpboards.com/attachments/discrete-mathematics-set-theory-logic-15/5623-laws-logic-q1-iv-1-png

I can see you are completely in the dark.

Before we continue is that a University course ??
 
solakis said:
I can see you are completely in the dark.

Before we continue is that a University course ??

Yep it is a uni course. It's Ok I have submitted the assignment. I have completed most of the q's so this one was just 2 points. Yes I am completely in the dark about this. I have a fraction of info from the textbook, study book and the lectures but doesn't actually explain how this concept actually works.
 
werebilby said:
Yep it is a uni course. It's Ok I have submitted the assignment. I have completed most of the q's so this one was just 2 points. Yes I am completely in the dark about this. I have a fraction of info from the textbook, study book and the lectures but doesn't actually explain how this concept actually works.

I am going to complete the problem for you and then you tell me what you do not understand

Working on the LHS of the relation we have:
[(p=>q)=>(p^r)]=

= [(~pvq)=>(p^r)]=......by implication on (p=>q)

= ~(~pvq)v (p^r)=......by implication on [(~pvq)=>(p^r)]

= (~~p^~q)v (p^r)=......by D.Morgan on ~(~pvq)

= (p^~q)v (p^r) =........ by D.Negation on ~~p

= p^(~qvr)=.........by distributive property

[Note: if you expand p^(~qvr) using the distributive property you will get (p^~q)v (p^r)]

=p^(q=>r).........by implication on (~qvr)

So starting from the LHS of the identity we ended on the RHS of the idendity.

This is a usual procedure when proving identities ,whether in real Algebra or propositional Algebra
 
Last edited:
Show that:

(p \to q) \to (p \to r) \;\equiv\;p \wedge (q \to r)
\begin{array}{ccccc}<br /> 1. &amp; (p \to q) \to (p \to r) &amp;&amp; 1 . &amp; \text{Given, LHS} \\<br /> 2. &amp; (\sim p \vee q) \to (\sim p \vee r) &amp;&amp; 2. &amp; \text{Implication} \\<br /> 3. &amp; \sim(\sim p \vee q) \vee (\sim p \vee r) &amp;&amp; 3. &amp; \text{Implication} \\<br /> 4. &amp; (p\: \wedge \sim q) \vee \sim p \vee r &amp;&amp; 4. &amp; \text{DeMorgan} \\<br /> 5. &amp; (p\:\vee \sim p) \wedge (\sim q\: \vee \sim p) \vee r &amp;&amp; 5.&amp; \text{Distributive} \\<br /> 6. &amp; t \wedge (\sim q \:\vee \sim p) \vee r &amp;&amp; 6. &amp; \text{Inverse} \\<br /> 7. &amp; \sim q \:\vee \sim p \vee r &amp;&amp; 7. &amp; \text{Identity} \\<br /> 8. &amp; \sim p \vee (\sim q \vee r) &amp;&amp; 8. &amp;\text{Comm. Assoc.} \\<br /> 9. &amp; \sim p \vee (q \to r) &amp;&amp; 9. &amp; \text{Implication} \\<br /> 10. &amp; p \to (q \to r) &amp;&amp; 10. &amp; \text{implication, RHS}<br /> \end{array}


 

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