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emeraldskye177
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Homework Statement
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Homework Equations
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The Attempt at a Solution
And I just don't know what to do from here... Any help will be greatly appreciated!
Hi, thanks for taking the time to respond. Does this look correct to you?Stephen Tashi said:I think you should use the distributive law on expressions like:
##(\lnot p \land \lnot q) \lor (q \lor \lnot r)##
to get:
## (\lnot p \lor ( q \lor \lnot r) ) \ \land \ (\lnot q \lor (q \lor \lnot r) )##
Then, in those propositions that involve only "##\lor##"'s, you can change the pattern ##A \lor B \lor C## to ## \lnot( \lnot A \land \lnot B) \lor C## and then get rid of the last ##\lor## by changing it to ##(\lnot A \land \lnot B) \implies C##.
emeraldskye177 said:Does this look correct to you?
The rules you are using change expressions to logically equivalent expressions, so your steps are guaranteed to result in a logical equivalence.Also, how would I prove that the original expression and the one I derived are equivalent using logical equivalencies?
Hi Stephen,Stephen Tashi said:You are doing the double negations like changing ##\lnot (\lnot p) ## to ##p## without showing it as a step. Some instructors may permit that.
The expression ##\lnot q \lor (q \lor \lnot r)## could be simplifed to ## ( \lnot q \lor q) \lor \lnot r## and then to ## T \lor \lnot r## and then to ##T##.The rules you are using change expressions to logically equivalent expressions, so your steps are guaranteed to result in a logical equivalence.
Of course, you can check your work by using a truth table.
If you had a rule that was not a logical equivalence such as ##p \lor q \lor r \implies p ## and you changed the expression ##(p \lor q \lor r)## to ##(p)## then you could not claim that such a step produced a new expression that was logically equivalent to the old expression. However, all the rules you listed use the relation ##\equiv##.
It might be simpler to continue by using the associative and commutative laws after you reach the expression:emeraldskye177 said:Can the original and derived expressions be further reduced?
Logical equivalencies are used to show that two statements are logically equivalent, meaning that they have the same truth value in all possible scenarios. This is important in mathematics because it allows us to simplify complex statements and make logical deductions.
There are several methods for proving logical equivalencies, including truth tables, algebraic manipulation, and logical rules. Truth tables involve systematically listing out all possible combinations of truth values for the statements and determining if they have the same truth value. Algebraic manipulation involves using known logical equivalencies to transform one statement into the other. Logical rules, such as the distributive property, can also be used to prove equivalencies.
Some common logical equivalencies include De Morgan's Laws, the commutative property, the associative property, and the distributive property. These equivalencies can be used to simplify complex statements and make logical deductions.
In computer science, logical equivalencies are used to simplify complex code and algorithms. They can also be used to optimize code and improve efficiency. Logical equivalencies are also important in troubleshooting and debugging code, as they can help identify and correct logical errors.
Logical equivalencies have many real-life applications, such as in legal arguments, scientific research, and critical thinking. They are also used in everyday decision-making, as we often use logical equivalencies to evaluate the validity of different options and make informed decisions.