Removing Implications Using Logic: Help Needed!

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In summary, this conversation is about a person's difficulty with an assignment that involves using laws of logic without a truth table. The person asks for help and another person provides a detailed explanation of how to approach the problem and arrives at the solution of (p->q)->(p->r)≡p∧(q->r). The conversation ends with the person thanking the other person and stating that they have submitted the assignment.
  • #1
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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  • #2
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
 
  • #3
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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  • #4
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??
 
  • #5
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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  • #6
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 ??
 
  • #7
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.
 
  • #8
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:
  • #9
Show that:

[tex](p \to q) \to (p \to r) \;\equiv\;p \wedge (q \to r)[/tex]

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


 

1. What is the process for removing implications using logic?

The process for removing implications using logic involves analyzing the given implications and applying logical rules to break them down into simpler statements. These statements are then rearranged or combined to create an equivalent statement without implications.

2. Why is it important to remove implications using logic?

Removing implications using logic allows for a clearer understanding of the given statements and helps to avoid any misunderstandings or misinterpretations. It also allows for easier manipulation and simplification of complex statements.

3. What are some common logical rules used to remove implications?

Some common logical rules used to remove implications include De Morgan's laws, distribution, and conditional elimination. These rules involve replacing implications with equivalent statements using logical operators such as AND, OR, and NOT.

4. Can removing implications change the meaning of a statement?

No, removing implications using logic does not change the meaning of a statement. It only rearranges or simplifies the statement to make it easier to understand and work with.

5. Are there any limitations to removing implications using logic?

Yes, there are certain limitations to removing implications using logic. It may not be possible to remove all implications from a given statement, and in some cases, the resulting statement may become more complex. It also requires a good understanding of logical rules and may be challenging for those who are not familiar with them.

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