What does it mean for something to imply something else in maths? I don't think I've grasped this concept and it's making understanding theorems and proofs really difficult for me. I know that a statement is a sentence which is either true or false, but not both. I think it might be the word 'imply' that's throwing me off. I've searched for definitions and examples, but they don't seem to make anything clearer for me. Thank you.
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$\begingroup$In most mathematical proofs, $P\implies Q \space \equiv\space \neg[P\land \neg Q]$.
There is also no causal relationship assumed between the antecedent $P$ and consequent $Q$, i.e. we do not assume that $P$ causes $Q$.
From the above definition we have:
- If we assume $P$ is true and can prove that $Q$ is also true, then we can infer that $P\implies Q$.
- If $P$ is true and $[P\implies Q]$ is true, then can infer that $Q$ is true.
$P \implies Q\space$ follows from any of the following:
- $\neg Q \implies \neg P$
- $\neg P \space \space$
- $Q \space \space $
A statement $A$ implies another statement $B$ (written as $A\Rightarrow B$), if from the truth of the former, it necessarily follows the truth of the latter.
Example. If I am in London, I am necessarily in England. So the statement "I am in London" implies the statement "I am in England".
On the other hand, if it is possible for $B$ to be true, while simultaneously $A$ is false, then $A$ is not implying $B$.
Example. If I am in England, I am not necessarily in London (I could be in Oxford). Hence, the statement "I am in England" does not imply the statement "I am in London".
$\endgroup$ $\begingroup$In math, the fact that a statement $A$ implies a statement $B$ is written this way: $A \implies B$
The meaning of $A \implies B$ is defined by this truth table:
$$ \begin{matrix} A & B & | & A \implies B\\ T & T & | &T\\ T & F & | &F\\ F & T & | &T\\ F & F & | &T\\ \end{matrix} $$
That means that $A \implies $B is false only when $A$ is true and $B$ is false. In the other cases $A \implies $B is true.
So let's say I have a theorem that states "If $n$ is a multiple of $6$ then it must be a multiple of $2$" ($n$ multiple of $6 \implies n$ multiple of $2$). For this theorem to be true, if you encounter a multiple of $6$ then it must be also a multiple of $2$ because if not the implication would be false. But if you encounter a number that is not a multiple of $6$ then it can be a multiple of $2$ or not without invalidating the theorem, because the theorem says nothing about numbers that are not multiple of $6$.
You can also see what the implication means by looking at sets:
If you take two statements $P$ and $Q$ then saying "$P$ implies $Q$" or equivalently $P \implies Q$ means that if $P$ holds then $Q$ holds. For instance, the following are some factual implications.
"$x$ is a real number greater than zero" implies "$(-1)\cdot x$ is a real number less than zero." This is true because if the first statement holds then one may conclude the second holds as well.
"x is an odd number" implies "There exists a natural number $k$ such that $x = 2k + 1$." Again, from the first statement one can conclude the second.
Here is an example of an incorrect implication:
- "x is a real number" implies "$x$ is less than zero." Here, one cannot conclude the second statement from the first, so it is incorrect to write this. What if $x = 0$?
Fundamentally, mathematics is a guide to reducing the unknown. When we have a statement "If $P$ then $Q$", that means if we somehow already know that $P$ is true, then we now can be assured that $Q$ is true. Each theorem that proves an implication allows us to expand our knowledge of true facts by using chains of implications. We start with known supposed true facts and, using chains of implications, we can conclude many other true facts on the basis of the suppositions. And finally arriving at what we want to prove to be true also. The Achilles' heel of this method is that we are dependant of the starting true facts, and if any one of them turns out to be false, then the results of our implications are now not assured to be true. Another weakness is that we could make several kinds of errors in applying the implications because "to err is human" and "a chain is as strong as its weakest link".
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