Duality discrete math problem. The dual of a compound proposition that contains only the logical operators ∨ , ∧ , and ¬ is the compound proposition obtained by replacing each ∨ by ∧ , each ∧ by ∨ , each T by F , and each F by T . The dual of s is denoted by s∗. Find the dual of these compound propositions.

Leila Jennings 2022-07-18 Answered
Duality discrete math problem
The dual of a compound proposition that contains only the logical operators ∨ , ∧ , and ¬ is the compound proposition obtained by replacing each ∨ by ∧ , each ∧ by ∨ , each T by F , and each F by T . The dual of s is denoted by s∗. Find the dual of these compound propositions.
a) p ¬ q
I got ¬ p q
b) p ( q ( r T ) )
My answer was ¬ p ( ¬ q r )
c) ( p ¬ q ) ( q F )
My answer was ( ¬ p q ) ¬ q
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Answers (2)

Tristan Pittman
Answered 2022-07-19 Author has 14 answers
Step 1
You did more as you should. Forming the dual just wants you to replace p by ¬ p for each literal p, ∨ by ∧ and vice versa and T by F. You did more than that, in dualising (2), one obtains
¬ p ( ¬ q ( ¬ r F ) )
Step 2
(you missed a ¬ in front of r). We have of course ¬ r F ¬ r, but this is not part of dualising. Same for (3), the dual proposition is
( ¬ p q ) ( ¬ q T )
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Karsyn Beltran
Answered 2022-07-20 Author has 5 answers
Step 1
What you have done is wrong. Why you have negated all the proposition and the then find the dual?
Like dual of ( p ¬ q )   i s   ( p ¬ q )   n o t   ( ¬ p q ) .
Here is the definition of dual of a compound proposition:
The dual of a compound proposition that contains only the logical operators ∨, ∧, and ¬ is the compound proposition obtained by replacing each ∨ by ∧, each ∧ by ∨, each T by F, and each F by T. The dual of s is denoted by s∗.
Step 2
Example: S = ( p q ) ( ¬ p q ) F
dual of S = ( p q ) ( ¬ p q ) T
So the correct answer to your question is:
a) p ¬ q
dual: p ¬ q
b) p ( q ( r T ) )
dual: p ( q ( r F ) )
c) ( p ¬ q ) ( q F )
dual: ( p ¬ q ) ( q T )
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