What is the simplified form of (p ∧ q) ↓ q using basic propositional logic?

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SUMMARY

The simplified form of the expression (p ∧ q) ↓ q is ¬q, as established through propositional logic. The discussion emphasizes the use of truth tables and the laws of logic to demonstrate this equivalence. The NOR operation, represented by (a ↓ b) = ¬(a ∨ b), is crucial in understanding the transformation of the expression. Participants in the forum provided guidance on how to approach the problem systematically.

PREREQUISITES
  • Understanding of propositional logic
  • Familiarity with truth tables
  • Knowledge of logical operations, specifically NOR
  • Basic skills in symbolic logic notation
NEXT STEPS
  • Study the construction and interpretation of truth tables in propositional logic
  • Learn about logical equivalences and their applications
  • Explore the properties and applications of the NOR operation in logic
  • Practice simplifying logical expressions using various logical laws
USEFUL FOR

This discussion is beneficial for students of logic, educators teaching propositional logic, and anyone seeking to deepen their understanding of logical expressions and simplifications.

moredumbimpossi
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Please help me with this thing. I'm so frustrated I can't understand propositional logic

Demonstrate this:

(p ∧ q) ↓ q ≡ ¬q

PLEASE.
 
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moredumbimpossi said:
Please help me with this thing. I'm so frustrated I can't understand propositional logic

Demonstrate this:

(p ∧ q) ↓ q ≡ ¬q

PLEASE.

Hi moredumbimpossi, welcome to MHB!

What have you tried? Where are you stuck?

Simplest method to prove something like this, is to set up a truth table.
Let's start with p=0 and q=0.
What is (p ∧ q) ↓ q = (0 ∧ 0) ↓ 0 then?
 
I like Serena said:
Hi moredumbimpossi, welcome to MHB!

What have you tried? Where are you stuck?

Simplest method to prove something like this, is to set up a truth table.
Let's start with p=0 and q=0.
What is (p ∧ q) ↓ q = (0 ∧ 0) ↓ 0 then?

Hi., thanks for replying
This has to be reduced with the laws of logic
 
moredumbimpossi said:
Hi., thanks for replying
This has to be reduced with the laws of logic

Oh, okay.
Let's first get to basic operations then.
In general, we have $(a ↓ b) = \lnot (a \lor b)$ don't we? It's a NOR after all.
What do we get if we replace the NOR (↓) in (p ∧ q) ↓ q by those basic operations?
 

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