Let p, q, and r be propositions and the expression r(p→q)→r be a contradiction. Then the expression (r→p)→q is


Question 20Answer

a.

Always true when p is false




b.

A tautology

c.

Always true when q is true


d.

A contradiction

e.

Always true when r is false

We are given that the expression \( r(p \rightarrow q) \rightarrow r \) is a contradiction. From this, we need to analyze the expression \( (r \rightarrow p) \rightarrow q \).

### Step 1: Analyze the given expression

The given expression \( r(p \rightarrow q) \rightarrow r \) is a contradiction. This means the truth table of the expression results in **false** for all possible truth values of \( p \), \( q \), and \( r \).

First, let's break down the expression \( r(p \rightarrow q) \rightarrow r \):

- \( p \rightarrow q \) is equivalent to \( \neg p \lor q \) (implication).
- Then \( r(p \rightarrow q) \) means \( r \land (\neg p \lor q) \), since conjunction takes place here.
- Finally, \( r(p \rightarrow q) \rightarrow r \) states that the result of \( r \land (\neg p \lor q) \) implies \( r \).

### Step 2: Examine \( (r \rightarrow p) \rightarrow q \)

Now, we examine the expression \( (r \rightarrow p) \rightarrow q \):

- \( r \rightarrow p \) is equivalent to \( \neg r \lor p \).
- So, \( (r \rightarrow p) \rightarrow q \) is \( (\neg r \lor p) \rightarrow q \), which is logically equivalent to \( \neg(\neg r \lor p) \lor q \), or simply \( (r \land \neg p) \lor q \).

### Step 3: Analyze the choices

Now let's go through the answer choices:

- **a. Always true when \( p \) is false**:
  If \( p \) is false, \( (r \land \neg p) \lor q \) becomes \( r \lor q \), which is not always true.

- **b. A tautology**:
  A tautology is true under all truth value assignments, but this is not the case for \( (r \land \neg p) \lor q \), as it is not always true.

- **c. Always true when \( q \) is true**:
  If \( q \) is true, the expression \( (r \land \neg p) \lor q \) is indeed always true, as \( q \) makes the whole expression true regardless of the values of \( r \) and \( p \).

- **d. A contradiction**:
  The expression is not always false, so it's not a contradiction.

- **e. Always true when \( r \) is false**:
  If \( r \) is false, the expression \( (r \land \neg p) \lor q \) simplifies to \( q \), which is not necessarily always true when \( r \) is false.

### Conclusion:

The correct answer is **c. Always true when \( q \) is true**.

Would you like further details or clarifications?

Here are 5 related questions you might consider:
1. What is the difference between a tautology and a contradiction in logic?
2. How can implication be rewritten in terms of disjunctions?
3. Can a complex logical expression be both a tautology and a contradiction under different conditions?
4. How do truth tables help in determining the validity of logical expressions?
5. What role does the truth value of \( q \) play in simplifying logical statements?

**Tip:** When analyzing logical expressions, breaking them into smaller parts and using truth tables often helps to clarify the relationships between propositions.

Let p, q, and r be propositions and the expression r(p→q)→r be a contradiction. Then the expression (r→p)→q is

Explore the logical analysis of the expressions r(p→q)→r and (r→p)→q. Understand how implications and contradictions work in propositional logic with detailed step-by-step solutions. Suitable for undergraduate-level students.

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