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 word "introduction" here is about logical introduction — bringing a new idea or assumption into a discussion — not the first paragraph of an essay.
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⟹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⟹Q as a proven statement.
Symbolically:
P⟹QP⋮Q
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 x2>4", you are being asked to perform a conditional introduction. You start by introducing the assumption x>2, work from there, and end with x2>4. Then you write the implication. …