Excerpt from a mathematical proof by contradiction involving discrete variables that serves to prove by contradiction that predicate P(n) is true for all nonzero integers n.
If n belongs to set C i.e. n >= c, then P(n) is false.
This implies the contrapositive,
If P(n) is true, n has to not be a part of set C i.e n < c.
Question is, why is the above implication immediately valid? Am I not assuming that n cannot take on 2 values simultaneously -- n being both >= c and < c, thus P(n) being both true and false simultaneously? What is the basis behind a predicate (or any proposition) being either true or false?
Also, if you don't mind, please do let me know if I am using some terminologies (i.e. discrete, predicate, contrapositive, assume) wrongly. I am extremely new to discrete mathematics and could really use some extra guidance.
Thank you.
Known: If $n \ge c$, then $P(n)$ is false.
Now let's assume that $P(n)$ is true. Then $n\ge c$ must be false. Otherwise, we know from above that $P(n)$ would be false. Thus, $n<c$. $\square$
I'm not seeing the connection to statistics, but that's how the contrapositive works.