Logical Induction

Question

"Prove that the number of people making an odd number of handshakes is always even." Though it can be easily proved by Induction but I wanted to ask is it correct to state the problem otherwise like "Even number of people always make odd number of handshakes."

Answer 1

The statement

"Even number of people always make odd number of handshakes."

is not logically equivalent to the statement

"the number of people making an odd number of handshakes is always even."

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The first statement can be written as:

$$\text{number of people is even}\implies \text{the number of handshakes is odd}$$

while the other can be written as

$$\text{the number of handshakes is odd}\implies\text{number of people is even}$$

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Logically, these two statements are obviously different, one being of the form $A\implies B$ while the other being $B\implies A$. To see how statements like this aren't always true, just set $A$ to be "person $x$ is a man", and $B$ to be "person $x$ is a human".

Clearly, the statement $A\implies B$ is true, but $B\implies A$ is not.

Answer 2

Your statement

Even number of people always make odd number of handshakes

is a bit weird and hard to understand. It seems to be saying: there is a specific group of people that is of even size, and they, no matter what, always make an odd number of handshakes. So an example of this would be if we had Alice, Bob, Charles, and Diana, and we would put them in various situations or occasions where they have to give handshakes, but no matter the occasion, these four people always end up giving an odd number of handshakes.

In addition, we could specify that Alice, Bob, Charles, and Diana are the only four people that have this knd of behavior: all other people make an even number of handshakes on at least some occasions.

Is this what you meant by your statement?

If so, it is not equivalent to the original statement, which says: no matter the occasion, the number of people giving an odd number of handshakes will always be even. The latter statement is mot making the claim that there is an even number of people that no matter the occasion, always give an odd number of handshakes; and it is certainly not claiming that the number of people that always give an odd number of handshakes is even.

If this is not what you are meaning ... then the only alternative reading I can make for your statement is exactly what the original statement would be saying: on any occasion, the number of people giving an odd number of handshakes is even. But then the statement would not just be equivalent, but simply be the same, just stated very awkwardly.