Summary of Discrete Math - 1.1.2 Implications Converse, Inverse, Contrapositive, and Biconditionals

Main Concepts:

Implications:
Defined as "if-then" statements (denoted as P → Q or "P implies Q"). The truth value of an implication can be tricky, with specific conditions determining when it is true or false.
Truth Table for Implications:
The truth values for P and Q are organized in a Truth Table. The Truth Table includes the following combinations:
- True (T), True (T) → True (T)
- True (T), False (F) → False (F)
- False (F), True (T) → True (T)
- False (F), False (F) → True (T)
The implication is false only when P is true and Q is false.
Converse, Inverse, and Contrapositive:
- Converse: Switch the order of P and Q (If Q then P).
- Inverse: Negate both P and Q (If not P then not Q).
- Contrapositive: Switch and negate (If not Q then not P). The Contrapositive always has the same truth value as the original implication.
Biconditionals:
Denoted as P ↔ Q (read as "P if and only if Q"). Both propositions must share the same truth value for the biconditional to be true. The Truth Table for Biconditionals shows that they are true when both propositions are either true or false.
Constructing Truth Tables:
The process of creating truth tables for Implications and Biconditionals is outlined. The tables help visualize the relationships and truth values of the propositions involved.

Creating a Truth Table:
- List all combinations of truth values for P and Q.
- For Implications, determine the truth values based on the conditions discussed.
- For Biconditionals, ensure both propositions have the same truth value.
Finding Converse, Inverse, and Contrapositive:
- Start by writing the implication in "if-then" form.
- Apply the definitions of Converse, Inverse, and Contrapositive to derive the new statements.