Q.In a restaurant 35 visitors can have lunch at a time. If the number of visitors increases by 7, then the expense of the restaurant on food increases by Rs. 42, while the average expenditure per head decreases by Rs. 1. Find the original expenditure of the restaurant.

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. …