# Binary Operation True or False statements

State True or False for each statment:
1. A binary operation on a set S assigns AT LEAST ONE element of S to each ordered pair of elements of S.

2. A binary operation on a set S assigns AT MOST ONE element of S to each ordered pair of elements of S.

3. A binary operation on a set S assigns EXACTLY ONE element of S to each ordered pair of elements of S.

4. A binary operation on a set S may assign more than one element of S to some ordered pair of elements of S.

I already know from the answer reference in the book I'm using, that numbers 1, 2, 3 are true and 4 is false. To me, the 3rd one being true makes perfect sense, no explanation needed. I know the definition of a binary operator. Also the 4th one being false makes 100% perfect sense to me. But I don't understand why the 1st and 2nd statements are true.

When reading the words "AT LEAST ONE" in the first statement, I interpret that the sentence opens the possibility of there existing 2 or 3 or 4, etc elements assigned. If this is the case, the statement would be false. Yet the books says it is true.

Same with the second statement. When reading the words "AT MOST ONE", I understand that the possibility exists that there could be NO element assigned because zero is less than one, or there could be one. If there is no element assigned, then it is false. But if there is one element assigned, it is true. The wording of the sentences confuse me.

Someone please explain this better for me!!! Thank you.

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Think about this: $$1 \ge 1 \\ 1 \le 1$$ Are these true or false?