Conditional Statements and Logical Forms

Conditional statements are foundational elements of logic and mathematics that establish a relationship between a hypothesis and a conclusion.
Original Statement:
- Symbolic Structure: $\text{If } A, \text{ then } B.$
- The hypothesis is represented by $A$ .
- The conclusion is represented by $B$ .
Inverse:
- Symbolic Structure: $\text{If not } A, \text{ then not } B.$
- Formed by negating both the hypothesis ( $A$ ) and the conclusion ( $B$ ) while maintaining their original positions.
Converse:
- Symbolic Structure: $\text{If } B, \text{ then } A.$
- Formed by swapping the positions of the hypothesis ( $A$ ) and the conclusion ( $B$ ).
Contrapositive:
- Symbolic Structure: $\text{If not } B, \text{ then not } A.$
- Formed by both swapping the positions of the hypothesis ( $A$ ) and conclusion ( $B$ ) and negating both.

Original Statement Example:
- Statement: If the evil mastermind executes her plan, then she will take over the world.
- Hypothesis ( $A$ ): The evil mastermind executes her plan.
- Conclusion ( $B$ ): She will take over the world.
Inverse Example:
- Statement: If the evil mastermind doesn't execute her plan, then she won't take over the world.
- Application: Negates the original hypothesis ( $\text{not } A$ ) and negates the original conclusion ( $\text{not } B$ ).
Converse Example:
- Statement: If the evil mastermind takes over the world, then she has executed her plan.
- Application: Reverses the roles of hypothesis ( $B$ ) and conclusion ( $A$ ).
Contrapositive Example:
- Statement: If the evil mastermind doesn't take over the world, then she hasn't executed her plan.
- Application: Reverses and negates both components ( $\text{not } B \rightarrow \text{not } A$ ).

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