Key Concepts and Formulas

• Tautology: A statement that is always true, regardless of the truth values of its components.
• Implication ($\Rightarrow$): p $\Rightarrow$ q is equivalent to $\sim$p $\vee$ q. It is only false when p is true and q is false.
• Disjunction ($\vee$): p $\vee$ q is true if either p is true, q is true, or both are true. It is only false when both p and q are false.
• Conjunction ($\wedge$): p $\wedge$ q is true only if both p and q are true. It is false otherwise.
• Negation ($\sim$): $\sim$p is true if p is false, and $\sim$p is false if p is true.

Step-by-Step Solution

Step 1: Rewrite the Implication

We are given the statement r $\vee$ ($\sim$p) $\Rightarrow$ (p $\wedge$ q) $\vee$ r. We want to find the value of 'r' such that the statement is a tautology. First, rewrite the implication using the equivalence p $\Rightarrow$ q $\equiv$ $\sim$p $\vee$ q.

So, r $\vee$ ($\sim$p) $\Rightarrow$ (p $\wedge$ q) $\vee$ r is equivalent to $\sim$(r $\vee$ ($\sim$p)) $\vee$ ((p $\wedge$ q) $\vee$ r).

Step 2: Apply De Morgan's Law

Apply De Morgan's Law to simplify the first term: $\sim$(r $\vee$ ($\sim$p)) $\equiv$ ($\sim$r) $\wedge$ p.

Therefore, the entire expression becomes: (($\sim$r) $\wedge$ p) $\vee$ ((p $\wedge$ q) $\vee$ r).

Step 3: Test the options

We will now test each of the given options for 'r' to see which makes the entire expression a tautology.

• Option (A): r = p

Substituting r = p, we get: (($\sim$p) $\wedge$ p) $\vee$ ((p $\wedge$ q) $\vee$ p).

Since ($\sim$p) $\wedge$ p is always false (a contradiction), it simplifies to: False $\vee$ ((p $\wedge$ q) $\vee$ p).

Now, (p $\wedge$ q) $\vee$ p is equivalent to p, since p $\wedge$ q can only be true if p is true. So the expression becomes: False $\vee$ p, which simplifies to p.

'p' is not a tautology, so option (A) is incorrect.

• Option (B): r = q

Substituting r = q, we get: (($\sim$q) $\wedge$ p) $\vee$ ((p $\wedge$ q) $\vee$ q).

Now, (p $\wedge$ q) $\vee$ q is equivalent to q, since p $\wedge$ q can only be true if q is true. So the expression becomes: (($\sim$q) $\wedge$ p) $\vee$ q.

This can be rewritten as: ($\sim$q $\vee$ q) $\wedge$ (p $\vee$ q).

Since $\sim$q $\vee$ q is always true (a tautology), the expression becomes: True $\wedge$ (p $\vee$ q), which simplifies to p $\vee$ q.

p $\vee$ q is not a tautology, so option (B) is incorrect.

• Option (C): r = $\sim$p

Substituting r = $\sim$p, we get: ($\sim$($\sim$p) $\wedge$ p) $\vee$ ((p $\wedge$ q) $\vee$ $\sim$p).

This simplifies to: (p $\wedge$ p) $\vee$ ((p $\wedge$ q) $\vee$ $\sim$p).

Further simplifying, we get: p $\vee$ ((p $\wedge$ q) $\vee$ $\sim$p).

Rearranging the terms: (p $\vee$ $\sim$p) $\vee$ (p $\wedge$ q).

Since p $\vee$ $\sim$p is always true (a tautology), the expression becomes: True $\vee$ (p $\wedge$ q).

This simplifies to True, which is a tautology.

So, option (C) is correct.

• Option (D): r = $\sim$q

Substituting r = $\sim$q, we get: ($\sim$($\sim$q) $\wedge$ p) $\vee$ ((p $\wedge$ q) $\vee$ $\sim$q).

This simplifies to: (q $\wedge$ p) $\vee$ ((p $\wedge$ q) $\vee$ $\sim$q).

Rearranging the terms: (p $\wedge$ q) $\vee$ $\sim$q.

This can be rewritten as: (p $\vee$ $\sim$q) $\wedge$ (q $\vee$ $\sim$q).

Since q $\vee$ $\sim$q is always true (a tautology), the expression becomes: (p $\vee$ $\sim$q) $\wedge$ True, which simplifies to p $\vee$ $\sim$q.

p $\vee$ $\sim$q is not a tautology, so option (D) is incorrect.

Step 4: Verify the Correct Answer

The correct answer is r = $\sim$p. Substituting this into the original expression: $\sim$p $\vee$ ($\sim$p) $\Rightarrow$ (p $\wedge$ q) $\vee$ $\sim$p

This simplifies to: $\sim$p $\Rightarrow$ (p $\wedge$ q) $\vee$ $\sim$p

Using the implication rule: $\sim$($\sim$p) $\vee$ ((p $\wedge$ q) $\vee$ $\sim$p)

Which simplifies to: p $\vee$ (p $\wedge$ q) $\vee$ $\sim$p

Rearranging: (p $\vee$ $\sim$p) $\vee$ (p $\wedge$ q)

Since p $\vee$ $\sim$p is always True: True $\vee$ (p $\wedge$ q)

This simplifies to True, which is a tautology.

Common Mistakes & Tips

• When simplifying logical expressions, remember De Morgan's Laws and the equivalences for implications.
• When testing options, try to simplify the expression as much as possible before substituting the value of 'r'.
• Remember that p $\vee$ True = True and p $\wedge$ False = False.

Summary

We are given a logical statement and asked to find the value of 'r' that makes the statement a tautology. We first rewrote the implication using its equivalent form with disjunction. Then, we substituted each of the given options for 'r' into the simplified expression and checked if the resulting expression was always true (a tautology). Only the option r = $\sim$p resulted in a tautology.

Final Answer

The final answer is \boxed{\sim p}, which corresponds to option (C).