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

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Discussion Overview

The discussion revolves around simplifying the expression (p ∧ q) ↓ q using basic propositional logic. Participants are exploring methods to demonstrate the equivalence of this expression to ¬q, focusing on logical laws and truth tables.

Discussion Character

  • Exploratory, Technical explanation, Homework-related

Main Points Raised

  • Some participants express frustration with understanding propositional logic and seek help in simplifying the expression.
  • One participant suggests using a truth table as a method to prove the equivalence of (p ∧ q) ↓ q and ¬q.
  • Another participant emphasizes the need to reduce the expression using the laws of logic rather than solely relying on a truth table.
  • A participant notes that the NOR operation can be expressed as ¬(a ∨ b), prompting a discussion on how to apply this to the given expression.

Areas of Agreement / Disagreement

Participants have not reached a consensus on the best method to simplify the expression, with differing opinions on the use of truth tables versus logical laws.

Contextual Notes

There are references to basic operations and logical laws, but the discussion does not resolve the steps needed to simplify the expression fully.

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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