if i study, then i will not fail mathematics
if i dont play , then i will study
but i failed amthematics
therefore i must have played
solve using truth table

The problem involves the use of propositional logic and requires constructing a truth table to solve it. Let's define the propositions first:

- \( P \): "I study."
- \( Q \): "I fail mathematics."
- \( R \): "I play."

The given conditions can be rewritten as logical statements:

1. **If I study, then I will not fail mathematics**: \( P \rightarrow \neg Q \)
2. **If I don't play, then I will study**: \( \neg R \rightarrow P \)
3. **I failed mathematics**: \( Q \)
4. **Therefore, I must have played**: \( R \) (this is the conclusion we're trying to prove).

### Steps:

We will construct a truth table based on these three propositions and determine when the premises are true and whether the conclusion (I played) follows logically.

### Propositions:

| \( P \) | \( Q \) | \( R \) | \( \neg Q \) | \( P \rightarrow \neg Q \) | \( \neg R \) | \( \neg R \rightarrow P \) | Premises | Conclusion \( R \) |
|--------|---------|--------|--------------|----------------------------|--------------|----------------------------|----------|------------------|
| T      | T       | T      | F            | F                          | F            | T                          | F        | T                |
| T      | T       | F      | F            | F                          | T            | T                          | F        | F                |
| T      | F       | T      | T            | T                          | F            | T                          | T        | T                |
| T      | F       | F      | T            | T                          | T            | T                          | T        | F                |
| F      | T       | T      | F            | T                          | F            | F                          | F        | T                |
| F      | T       | F      | F            | T                          | T            | F                          | F        | F                |
| F      | F       | T      | T            | T                          | F            | F                          | T        | T                |
| F      | F       | F      | T            | T                          | T            | F                          | T        | F                |

### Explanation:

- The premises \( P \rightarrow \neg Q \), \( \neg R \rightarrow P \), and \( Q \) need to be true to consider the scenario valid.
- Look for the rows where these premises hold.
- We can see from the table that the only row where the premises are true (T) is when \( P = F \), \( Q = T \), and \( R = T \), which means I **failed** mathematics and I **played**.

### Conclusion:

Since the truth table shows that when the premises are true, \( R = T \) (i.e., "I played"), the conclusion is correct: **I must have played**.

---

Do you want further details or have any questions? Here are 5 related questions:

1. How is the truth table method useful in proving logical statements?
2. What happens if the premises are false?
3. How can we relate this to real-world decision-making?
4. What if additional conditions were added to the problem? How would the truth table change?
5. Can we draw the same conclusion without using a truth table?

**Tip**: When constructing truth tables, always start by identifying and organizing the key propositions.

if i study, then i will not fail mathematics if i dont play , then i will study but i failed mathematics therefore i must have played solve using truth table

Solve a propositional logic problem using truth tables. The problem involves logical implications like 'If I study, then I will not fail mathematics' and 'If I don't play, then I will study'. This solution covers logical statements, truth tables, and theorems suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation