Introduction — What It Means and Why It Matters

You've been solving problems your whole life. When you see a new type of question, you first figure out what it's asking, then decide which tools to use, then carry out the steps. That first step — figuring out what's going on — is the heart of an introduction in mathematics and science.

An introduction is not a summary. It's a setup. It tells you the players, the rules, and the goal before the action starts.

The Intuition

Imagine you walk into a room and see a chessboard with pieces already arranged. Without knowing whose turn it is, what the last move was, or whether you're playing standard chess or a variant, you can't make sense of anything. An introduction is like being told: "White to move, standard rules, checkmate in two."

In a proof, an introduction declares the assumptions you're allowed to use. In a problem, it states the given data and the unknown you're after. In a definition, it names the new object and lists its essential properties.

The Precise Statement

In formal logic and mathematics, an introduction rule tells you how to legitimately bring a new statement into a derivation. The most common example is the conditional introduction (also called "direct proof" or "→-introduction"):

If from assumption $P$ you can derive $Q$, then you may conclude $P \implies Q$.

That is, to prove "if $P$ then $Q$", you temporarily assume $P$ is true, work through the reasoning, and arrive at $Q$. Then you discharge the assumption and write $P \implies Q$ as a proven statement.

Symbolically:

The line means "from the top, infer the bottom." The dots represent a valid chain of reasoning.

Why This Matters for Exams

When you see a question that asks you to "prove that if $x > 2$ then $x^2 > 4$", you are being asked to perform a conditional introduction. You start by introducing the assumption $x > 2$, work from there, and end with $x^2 > 4$. Then you write the implication.

Without understanding introduction, students often try to prove the implication directly without assuming the hypothesis — which is impossible. The introduction rule tells you exactly how to begin.

A Concrete Example

Problem: Prove that if $n$ is an even integer, then $n^2$ is even.

Solution using introduction:

1. Introduce the assumption: Assume $n$ is even.
2. Work from it: By definition, $n = 2k$ for some integer $k$. Then $n^2 = (2k)^2 = 4k^2 = 2(2k^2)$, which is even.
3. Conclude the implication: Therefore, if $n$ is even, then $n^2$ is even.

The introduction happened in step 1. Without it, you'd have no starting point.

The Big Picture

Every time you start a proof with "Suppose ..." or "Let ...", you are making an introduction. It's the logical equivalent of saying "Let's play this game with these rules." The rest of the proof is just following those rules to the conclusion.

Master introduction, and you master the first and most critical step of any proof.