Consider the following theorem: “If n is an even integer, then n + 1 is odd.” Give a Proof by Contradiction of this theorem.
even means a zero remainder when divided by 2
so, what if you add 1?
what happens when you divide by 2?
Ah, the joy of proofs! Alright, brace yourself for a wild ride filled with logic and laughter.
Let's assume that n is an even integer, but n + 1 is NOT odd. So what could it be? Well, there are two possibilities: either it's another even number, or it's some sort of craziness we've never seen before. Let's examine both cases:
Case 1: n + 1 is another even number. If that's true, we can say that n + 1 = 2k, where k is an integer. But hold on a second, if we have an even number, we can rewrite it as 2k - 1, where k is still an integer. And there we have it! n + 1 equals both 2k and 2k - 1 at the same time. But that's like saying one plus one equals two and one plus one equals three – it just doesn't add up, literally!
Case 2: n + 1 is some sort of craziness we've never seen before. If that's the case, well, we're in deep trouble here because we can't really prove or disprove something that doesn't make sense. Plus, mathematics already has enough challenges; we don't need imaginary numbers playing tricks on us!
Since both possibilities lead us down paths of insanity, we conclude that our assumption was wrong! Therefore, if n is an even integer, n + 1 must be an odd number. QED! (And no, QED doesn't stand for "Quite Exquisitely Dewy," although that would be a pretty awesome math term.)
So, my friend, let this silly proof bring you joy and laughter while you navigate the sometimes mind-bending world of mathematics!
To prove the theorem "If n is an even integer, then n + 1 is odd" by contradiction, we assume the opposite statement: "If n is an even integer, then n + 1 is not odd." This means we assume that n + 1 is even.
Step 1: Assume n is an even integer.
Step 2: Assume n + 1 is even.
Step 3: If n is even, then there exists an integer k such that n = 2k, where k is also an integer. This is the definition of an even number.
Step 4: If n + 1 is even, then there exists an integer m such that n + 1 = 2m, where m is also an integer. This is the definition of an even number.
Step 5: Now, let's substitute the expression for n from Step 3 into the equation in Step 4: 2k + 1 = 2m.
Step 6: Simplifying Step 5, we get 2k + 1 = 2m.
Step 7: If we subtract 2k from both sides of the equation in Step 6, we get 1 = 2m - 2k.
Step 8: Further simplifying Step 7, we get 1 = 2(m - k).
Step 9: Since m - k is an integer (the difference of two integers is an integer), we can rewrite Step 8 as 1 = 2q, where q = (m - k) is an integer.
Step 10: It is well-known that there is no integer solution to the equation 1 = 2q, as the right-hand side is always even, while the left-hand side is always odd.
Step 11: Since Step 10 contradicts our assumption in Step 2, the assumption that n + 1 is even, we can conclude that the opposite statement is false.
Step 12: Therefore, if n is an even integer, then n + 1 is odd. QED (quod erat demonstrandum), which means "the proof is complete."
To prove the theorem "If n is an even integer, then n + 1 is odd" by contradiction, we assume the opposite of the statement and show that it leads to a contradiction. Here's how we can do it:
Proof by Contradiction:
1. Assume that n is an even integer such that n + 1 is not odd.
2. If n + 1 is not odd, then it must be even. Therefore, we can write n + 1 = 2k for some integer k.
3. Rearranging the equation, we get n = 2k - 1.
4. Since n is an even integer, it can be written as n = 2m for some integer m. Substituting this into the equation, we have 2m = 2k - 1.
5. By subtracting 2k from both sides, we get 2m - 2k = -1.
6. Factoring out 2, we have 2(m - k) = -1.
7. Since m - k is an integer, we see that -1 must be divisible by 2, which leads to a contradiction.
8. Hence, our assumption that n + 1 is not odd must be incorrect, and therefore, if n is an even integer, then n + 1 is odd.
In this proof by contradiction, we assume the opposite of the statement and show that it leads to a contradiction. Since a contradiction cannot occur, we conclude that the original statement is true.