At its heart, mathematics is a special kind of language designed for precision. It avoids the fuzzy meanings we often find in everyday speech. The basic sentence in this language is called a proposition. A proposition is a statement that can be definitively labeled as either true or false, with no grey area in between.

Propositions are the building blocks. We combine them using logical connectives to form more complex ideas. Think of them as the conjunctions of mathematics. Let's represent two simple propositions with the letters $P$ and $Q$.

• AND ($∧$): $P ∧ Q$ is true only if both $P$ and $Q$ are true. For example, "It is raining AND it is cold."
• OR ($∨$): $P ∨ Q$ is true if at least one of them is true. "I will have coffee OR I will have tea." This is true if you have coffee, tea, or both.
• NOT ($¬$): $¬P$ simply flips the truth value of $P$. If $P$ is true, $¬P$ is false. For example, if $P$ is "The dog is barking," then $¬P$ is "The dog is not barking."
• IMPLIES ($→$): This is the classic "if-then" statement. $P → Q$ means "If $P$ is true, then $Q$ must be true." Think of it as a promise. If I say, "If it rains, then I will bring an umbrella," the only way I've broken my promise is if it rains and I don't bring an umbrella.

How do we keep track of what's true and false when propositions get complicated? We use a tool called a truth table. It's a simple chart that shows the outcome for every possible combination of truth values.

The table for AND shows that the combined statement is only true when both parts are true. Now look at the table for OR.

As long as at least one part is true, the OR statement holds up. The most interesting table is for IMPLIES ($P → Q$). Remember the promise: "If $P$, then $Q$."

The first two rows make sense. If I make the promise and the condition is met ($P$ is true), I must follow through ($Q$ must be true) for the promise to be unbroken. The second row shows the broken promise.

The last two rows can feel strange. If the condition isn't met ($P$ is false), the promise hasn't been broken, regardless of what happens with $Q$. If it doesn't rain, my promise about bringing an umbrella is still intact whether I bring one or not. So, when the premise is false, the implication is considered true.

Another fundamental concept is the set. A set is just a collection of distinct items. The items, called elements, can be anything: numbers, letters, people, or even other sets. We usually write sets using curly braces.

For example, the set of primary colors is {red, yellow, blue}. The set of the first five positive even numbers is {2, 4, 6, 8, 10}.

We can perform operations on sets, much like we do with numbers. Two of the most common are union and intersection.

So if Set A = {1, 2, 3} and Set B = {3, 4, 5}:

• A ∪ B = {1, 2, 3, 4, 5}
• A ∩ B = {3}

Logic and sets provide the language and materials for mathematics. But how do we build with them? We need rules. In mathematics, the entire structure is built upon a foundation of axioms.

You can't prove an axiom. You just have to accept it as a fundamental rule of the game you're playing. For example, a basic axiom of arithmetic is that $a + b = b + a$. We don't prove it; we assume it's true and build from there.

Once we have axioms, we can start proving other statements, which are called theorems. A theorem is a proposition that has been shown to be true through a rigorous, step-by-step logical argument called a proof. Each step in a proof must follow logically from the axioms, definitions, or previously proven theorems.

This process—starting with axioms, then using logic to prove theorems—is the core of mathematical reasoning. It's how we move from simple, self-evident truths to complex and often surprising conclusions, all while being certain that every step along the way is sound.

Let's review the key terms we've introduced before trying a few questions.

Ready to test your understanding?

Understanding these basic elements—propositions, sets, and the axiom-theorem structure—gives you a framework for how all of mathematics is built and communicated.