Discrete Math - Show Logically $X$ beat $Y$ exactly 2 Times

Question

I was given a statement that I need to show logically. The question is #6 on this page. I have figured out the other 3 on my own but am stuck on this part of question:

Define the predicate $B(x, y)$ to mean that $x$ beat $y$ in the race. Give the logical expression equivalent to 'Sam beat exactly two people'.

I know that the use of a universal or existential quantifier is most likely required here but I am not sure how to write the expression "equal to two" in conjunction with them.

Any help would be greatly appreciated.

Answer 1

HINT: You want to say that there exist people $x$ and $y$ such that:

• Sam beat $x$;
• Sam beat $y$;
• $x$ and $y$ are not the same person; and
• for any person $z$, if Sam beat $z$, then $z$ is $x$ or $z$ is $y$.