1) a) Decide if (q  ( p~ q))~ p is a tautology using propositional equivalence and the

laws of logic.
b) Prove that the following argument is valid:
q
q r
p r
p
~
_________
~



c) Replace p, q and r in 1b) by statements and construct an argument.

No clue how to even start. Thank you

Many of the symbols aren't showing up. Try rewriting using ∧ and ∨, or & and |.

To determine if a statement is a tautology, we need to use the laws of logic and propositional equivalences to simplify and analyze the given statement. Let's break down each part of the question and work through them step by step.

a) Decide if (q ∧ (p → ¬q)) → ¬p is a tautology using propositional equivalence and the laws of logic.

To determine if (q ∧ (p → ¬q)) → ¬p is a tautology, we can follow these steps:

Step 1: Simplify the expression using logical laws and propositional equivalences.
First, let's simplify the statement p → ¬q using the definition of implication. The implication p → ¬q is equivalent to ¬p ∨ ¬q.

Now, substitute the simplified expression into our original statement to get (q ∧ (¬p ∨ ¬q)) → ¬p.

Step 2: Continue simplifying using logical laws and propositional equivalences.
To simplify (q ∧ (¬p ∨ ¬q)), we can distribute the conjunction operator ∧ over the disjunction operator ∨. This gives us (q ∧ ¬p) ∨ (q ∧ ¬q), which further simplifies to (q ∧ ¬p) ∨ False. Since (q ∧ ¬p) always evaluates to False when q = False or p = True, we can replace (q ∧ ¬p) ∨ False with False.

Now our statement becomes False → ¬p.

Step 3: Analyze the final expression.
In propositional logic, False → p is always True. Therefore, False → ¬p is always True as well.

Since the final expression is always True, we can conclude that (q ∧ (p → ¬q)) → ¬p is a tautology.

b) Prove that the following argument is valid:
q
q → r
p → r
p
~
_________
~
∴ → ∴

To determine if this argument is valid, we can use deductive reasoning and logical laws to show that if all the premises are true, then the conclusion must also be true.

Step 1: Examine the premises and the conclusion.
The premises are q, q → r, p → r, and p.
The conclusion is ¬.

Step 2: Use logical laws to simplify the premises and find a contradiction.
Since the premises contain implications, we can break them down using the definition of implication:

q → r is equivalent to ¬q ∨ r.
p → r is equivalent to ¬p ∨ r.

Using these equivalences, the premises can be rewritten as:
Premise 1: q
Premise 2: ¬q ∨ r
Premise 3: ¬p ∨ r
Premise 4: p

Step 3: Apply logical laws to derive a contradiction.

Based on the premises, we can follow these steps:

From Premise 1 (q) and Premise 2 (¬q ∨ r), we can deduce r using the rule of disjunction elimination.

From Premise 4 (p) and Premise 3 (¬p ∨ r), we can deduce r using the rule of disjunction elimination.

Now, we have r and ¬r, which is a contradiction.

Step 4: Conclude that the argument is valid.
Since we obtained a contradiction by assuming all the premises are true, we can conclude that the given argument is valid.

c) Replace p, q, and r in 1b) by statements and construct an argument.

To replace p, q, and r with statements and construct an argument, we need the specific statements you want to use. Please provide the statements, and I'll be happy to help you construct the argument.