Have troubles with contradiction of a statement

Question

"Suppose that there are 13 people in a room. Prove: "At least two of these people were born in the same month". Use the indirect method."

The question I have is: Which of the following (if any) are contradictions of "At least two of these people were born in the same month"

1. Less than two of these people were born in the same month.
2. Less than two of these people were born in different months.
3. All people were born in different months.

Should I first parapharase the statement to be proved in "if-then" form?

Answer 1

Let $X_1, X_2, \dots, X_{12}$ be variables where $X_i$ represents the number of people in your group born in the $i^{th}$ month.

The phrase "at least two people are born in the same month" can be rephrased using the above variables as "$\exists i$ s.t. $X_i\geq 2$.

To negate the phrase, remember that $\exists$ get replaced by $\forall$ and vice versa, and $\geq$ gets replaced by $<$ (and more rules such as demorgans which doesn't apply here).

Thus, the negation of the phrase is $\forall i: X_i<2$, I.e. "All months has strictly fewer than two people born in it", or equivalently phrased as "All people are born in different months."

---

This example is a classic use of the pigeonhole principle. Suppose that the first twelve people are born in different months (else we would already be done). What can be said about the thirteenth person? (remember, there are only twelve months in the year)

Answer 2

I would translate the phrase "At least two of these people were born in the same month" as, "There exists at least one pair of distinct people that were born in the same month".

Then the negation of this statement would be "There does not exist a pair of distinct people born in the same month", in other words, your statement that all people were born in different months.

To address what you wrote specifically:

Less than two of these people were born in the same month.

This statement is a bit odd, because if you're referring to "less than two people", you must be referring to either $0$ or $1$ person. Then it's unusual to make statements about $0$ or $1$ person being born in the same month. Typically when we say "born in the same month", we're referring to a collection of at least two people, usually all distinct (and most often, we use "born in the same month" to talk about exactly two distinct people).

Less than two of these people were born in different months.

The same applies here, linguistically. Also, this is a bit ambiguous. On the one hand, I could take it to mean "Everyone was born in the same month". Or is it saying that, most of the people are born in the same month (say January), and either $0$ or $1$ person wasn't born in January?

All people were born in different months.

Bingo, and my point above explains one way to see this.