There are some coins in a straight line, all showing heads. You can choose any coin to flip. When you flip one, your friend must flip the coins directly next to it (the ones on its left and right, if there are any). For example: if you flip the first coin, your friend only flips the second coin, and if you flip the second coin, your friend flips the first and third coins.
The question is: no matter how many coins there are, can we always make all of them show tails in the end?
The product of two numbers \(a\) and \(b\) is greater than \(100\). Prove that at least one of the numbers is greater than \(10\).
Often in maths we want to prove statements of the form “If A, then B.” For example: “If a number is divisible by \(4\), then its even". Sometimes it’s tricky to prove such statements directly — that is, by starting with “suppose \(A\) is true” and trying to show “\(B\) must also be true.”
Luckily, there’s another way! A statement of the form “If A, then B” means exactly the same thing as “If not B, then not A.” This second way of writing it is called the contrapositive. Here’s an everyday example: “If it rains, then I take my umbrella". Is exactly the same as saying “If I didn’t take my umbrella, then it’s not raining".
When we use this method in maths, we often say we’re proving by contrapositive: instead of proving “If A then B”, we prove “If not B then not A.”
We sometimes write “If \(A\) then \(B\)” as \(A \implies B\), which is pronounced “\(A\) implies \(B\)”, and its contrapositive is: \(\text{not }B \implies \text{not }A.\) This way of thinking often makes a proof much simpler! Let’s see some examples to learn how to use this method.
What is the contrapositive of the statement “If my car won’t start, then the battery is flat or there is no fuel”?
What is the contrapositive of the statement: ”If the temperature is above \(40^\circ\), then it is hot and sunny."
Some lines are drawn on a large sheet of paper so that all of them meet at one point. Show that if there are at least \(10\) lines, then there must be two lines whose angle between them is at most \(18^\circ\).
A whole number \(n\) has the property that when you multiply it by \(3\) and then add \(2\), the result is odd. Use a proof by contrapositive to show that \(n\) itself must be odd.