Suppose \(a>b\) and \(c>d\). Prove that \(a+c>b+d\).
Using mathematical induction prove that \(2^n \geq n + 1\) for all natural numbers.
Circles and lines are drawn on the plane. They divide the plane into non-intersecting regions, see the picture below.
Show that it is possible to colour the regions with two colours in such a way that no two regions sharing some length of border are the same colour.
Consider a number consisting of \(3^n\) digits, all ones, such as 111 or 111111111 for example. Show that such a number with \(3^n\) digits is divisible by \(3^n\).
Numbers \(1,2,\dots,n\) are written on a whiteboard. In one go Louise is allowed to wipe out any two numbers \(a\) and \(b\), and write their sum \(a+b\) instead. Louise enjoys erasing the numbers, and continues the procedure until only one number is left on the whiteboard. What number is it? What if instead of \(a+b\) she writes \(a+b-1\)?
Prove that
(a) \[1^2 + 2^2 + 3^2 + \dots + n^2 = \frac{1}{6} n (n+1)(2n+1)\]
(b) \[1^2 + 3^2 + 5^2 + \dots + (2n-1)^2 = \frac{1}{3} n (2n-1)(2n+1)\].
Is “If you come here, then you are mad” the same thing as “If you are not mad, then you wouldn’t have come here”.
Is “if \(x = y\) then \(x^2= y^2\)” the same thing as “if \(x^2 \neq y^2\) then \(x \neq y\)”?
What is common between the two examples above? In fact, if you want to know some fancy words (you should understand what they mean, of course), we just stated that a direct proof and a proof by contrapositive is the same thing. In simple words it means that “If A then B” is the same thing as “If not B, then not A”.
A proof by contrapositive can be very useful. In some problems it is much easier to prove “If not B, then not A” compare to “If A then B”. Let’s consider another example, where a proof by contrapositive can be very useful
There are 10 lines drawn on the plane, all intersecting at the same point. Show that there will be at least two lines with angle between them less than \(18^o\).
Is “If you are not mad, then you growl when you are angry and wag your tail when you are pleased” the same thing as “If you don’t growl when you are angry or don’t wag your tail when you are pleased, then you are mad”?