Discrete Maths — Introduction to Propositional Logic + 6 more topics

Propositional logic focuses on propositions, which are declarative sentences that are unambiguously either true (T) or false (F), but not both. Questions, commands, or subjective opinions aren't propositions.

For example:
* "The sky is blue." (True)
* "2 + 2 = 5." (False)
* "Go clean your room!" (Not a proposition)

We use lowercase letters (p, q, r, s, etc.) to represent propositions.
* Let p be "It is raining."
* Let q be "The sun is shining."

In discrete math, we often work with propositions, which are declarative sentences that are either true or false, but not both. For example, "The sky is blue" is a proposition. "Is the sky blue?" is not.

We use logical operators to combine or modify these propositions, creating more complex statements. Let's dive into two basic ones: negation and conjunction.

Negation is the simplest operator. It takes a single proposition and reverses its truth value. If a statement P is true, its negation not P is false. If P is false, not P is true.

Notice the only time p → q is false is when p is true and q is false. This is the "broken promise" scenario.

1. Converse: q → p (If q, then p) - Swaps hypothesis and conclusion.
2. Inverse: ¬p → ¬q (If not p, then not q) - Negates both hypothesis and conclusion.
3. Contrapositive: ¬q → ¬p (If not q, then not p) - Swaps and negates both.

In discrete mathematics, we often combine statements using logical operators. Disjunction and Exclusive Or are two key operators, and while they sound similar, they have distinct meanings and applications.

Disjunction, denoted by the symbol $\lor$ (read as "or"), is a logical operation that returns true if at least one of the statements it connects is true. This means if statement A is true, or statement B is true, or both A and B are true, the disjunction A $\lor$ B is true. The only time A $\lor$ B is false is when both A and B are false.

A biconditional statement connects two propositions, let's call them P and Q, using the phrase "if and only if" (often shortened to "iff"). We write it symbolically as P ↔ Q.

What P ↔ Q really means is two things:
1. P implies Q (P → Q)
2. Q implies P (Q → P)

Both of these implications must be true for the biconditional (P ↔ Q) to be true.