City population N and M club, any club membership is an even number of members of both clubs prove m <n

Are the clubs mutually exclusive?

yes

To prove that M < N, we need to show that the club membership for M is less than the club membership for N, where both club memberships are even numbers.

To start, let's assume that M is greater than or equal to N, and both M and N represent the club memberships. Since both M and N are even numbers, we can express them as M = 2x and N = 2y, where x and y are positive integers.

Now, if we assume that M is greater than or equal to N, we can write this as 2x ≥ 2y. Dividing both sides of the inequality by 2, we get x ≥ y.

Since x and y are positive integers, if x is greater than or equal to y, it means there must be more or an equal number of club members in M compared to N. However, we know that N represents the city population, which means the number of people in the city should be greater than the number of people in any given club.

This contradiction shows that our initial assumption was incorrect, and M cannot be greater than or equal to N. Therefore, we can conclude that M < N, i.e., the club membership for M is less than the club membership for N.