b) Show by the truth that the following statements are logically equivalent:
Statement 1: Rich men are unhappy
Statement 2: Men are unhappy or poor (Let ‘poor’ indicate ‘not rich’)
To show that the statements are logically equivalent, we need to show that if one statement is true, then the other statement must also be true, and vice versa.
Let's consider the truth values of both statements:
Statement 1: Rich men are unhappy.
If we have a counterexample where there exists a rich man who is happy, then Statement 1 is false. However, if all rich men are unhappy, then Statement 1 is true.
Statement 2: Men are unhappy or poor.
If there exists a man who is happy, then Statement 2 is false. However, if all men are unhappy or poor, then Statement 2 is true.
To show logical equivalence, we need to check if the truth values of the statements match in all possible cases. Let's construct a truth table to evaluate all possibilities:
| Rich | Happy | Poor | Statement 1: Rich men are unhappy | Statement 2: Men are unhappy or poor |
|------|-------|------|---------------------------------|------------------------------------|
| 0 | 0 | 0 | True | True |
| 0 | 0 | 1 | True | True |
| 0 | 1 | 0 | False | False |
| 0 | 1 | 1 | True | True |
| 1 | 0 | 0 | True | True |
| 1 | 0 | 1 | True | True |
| 1 | 1 | 0 | False | False |
| 1 | 1 | 1 | True | True |
From the truth table, we observe that the truth values of both statements are identical in all cases. Therefore, we can conclude that Statement 1: "Rich men are unhappy" and Statement 2: "Men are unhappy or poor" are logically equivalent.