Задача 7.

Find mistakes in the following proofs: a) A proof of $n>n+1$. Let this statement be true for $n$, i.e. $n>n+1$. By adding $1$ to the both sides of the inequality, we get $(n+1)>(n+1)+1$. Thus, the inequality is true for $n+1$. b) A proof that in a herd of $N$ cows all the cows have the same color. The induction base. In a herd of one cow, all the cows have the same color, obviously. The induction step. Suppose that in any herd of $N$ cows all the cows have the same color. Now we show that in any herd of $N+1$ cows all the cows have the same color. Consider a herd of $N+1$ cows. Choose a cow and call it A. The remaining $N$ cows have the same color. Choose another cow from the herd of $N+1$ and call it B. The remaining $N$ cows have the same color. In particular, A has identical color as the other cows, except B. And B has the same identical color as the other cows, except A, see the picture below. Thus, A, B and all the other cows have identical color. c) There are several cities in a country. Some pairs of cities are connected by roads and any city is connected with at least one other. We show that we can drive from any city to any other. Apply mathematical induction using the number of cities. The induction base is a country with only one city. Now we proof the induction step. Consider a country with $n$ cities and add one new city to it. We can drive between the old $n$ cities, it remains to show that from the new city we can drive to any old city. Given that any city connected with at least one other, the new city is connected with at least one old. So, we can drive to the old city and from there drive to any other city. Thus, we proved the induction step.