Direct proof Definition and 18 Threads

It's very easy to prove the area formula for a rectangle when both length and width are positive integers, but I cannot prove it when length or width or both are rational or irrational numbers. I need an intuitive proof that is as simple as possible without using very advanced math like calculus.

Summary:: . When asked to prove by Induction, I'm asked to prove a statement of the form: Prove that for all natural numbers ##n##, ## P(n) ## Which means to prove: ## \forall n ( P(n) ) ## ( suppose the universe of discourse is all the natural numbers ) Then, I see people translating...

Homework Statement 1. Show that for all real numbers x and y: a) |x-y| ≤ |x| + |y| Homework Equations Possibly -|x| ≤ x ≤ |x|, and -|y| ≤ y ≤ |y|? The Attempt at a Solution I tried using a direct proof here, but I keep getting stuck, especially since this is my first time ever coming...

Okay, these are my last questions and then I'll get out of your hair for a while. For 1, I have already done a proof by contradiction, but I'm supposed to also do a direct proof. Seems like it should be simple? For 2, this seems obvious because it's the definition of an integral. My delta is...

So the definition of a bounded sequence is this: A sequence ##(x_{n})## of real numbers is bounded if there exists a real number ##M>0## such that ##|x_{n}|\le M## for each ##n## My question is pretty simple. How does one choose the M, based on the sequence in order to arrive at the...

Hello, I'm having trouble with an assigned problem, not really sure where to begin with it: Prove that if $$a \in R$$ and $$b \in R$$ such that $$0 < b < a$$, then $${a}^{n} - {b}^{n} \le {na}^{n-1}(a-b)$$, where n is a positive integer, using a direct proof. Pointers or the whole proof would...

Homework Statement Hypotheses: not a, b or not c, b→ (a and d), e→(c) Conclusion: not e 2. The attempt at a solution: So far, I have this: 1) not a as premise 2) b or not c as premise 3) b→ (a and d) as premise 4) e→(c) as premise 5) a by Step 1 and Law of Excluded Middle. 6) c is true...

Prove the following theorem: Theorem For a prime number p and integer i, if 0 < i < p then p!/[(p− i)! * i] * 1/p Not sure how to go about this. I wanted to do a direct proof and this is what I've got so far. let i = p-n then p!/[(p-n)!*(p-n)] but that doesn't exactly prove much.

Ok, so it's very easy to show P v P = P (where = is logically equivalent) using a truth table as well as using a conditional proof. P v P Premise ~p Assumption p Disjunctive Syllogism (1, 2) p & ~p Conjunction (3, 4) ~p --> (p & ~p) Conditional Proof (2--4)...

Statements: x is an integer x is a prime number if x doesn't consist of any prime factors ≤√x Proof: Since (√x + 1) * (√x + 1) > √x * √x x must be a prime Questions: Whould you consider this a non-rigorous direct proof? If not, what does it lack? Is this a good approach trying to...

Hi all, I am trying to proof the following question. If a is an integer, divisible by 4, then a is the difference of two perfect squares now by the definition of divisibility if 4 divides a then there is a natural number k such that a = 4k Can someone how should I do it with direct proof by...

Homework Statement im supposed to use a direct proof to prove that if 1-n^2>0 then 3n-2 is odd for all n∈Z Homework Equations The Attempt at a Solution if you let n∈z then suppose that 1-n^2>0 then 1>n^2 but the only inter n such that 1>n^2 is 0. 3x0-2=-2 as -2=2(-1), -2 is even ...

Let n be an integer. Prove that if n + 5 is odd, then 3n + 2 is even. So the instructions say to use a direct proof. I couldn't figure that method out, so I used a controposition proof and that seemed to work ok. Here are my contraposition steps: Assume 3n+2 is odd Def of odd: n=2k+1...

"A direct proof is a proof in which the truth of the premises of a theorem are shown to directly imply the truth of the theorem's conclusion." Here are the premises: (P -> R) ^ (Q -> S) ^ (~P) ^ (P v Q) and the conclusion: (S v R) ^ (~P) Now what I do not understand why we are...

Homework Statement Prove that for all integers n and m, if n-m is even then n3-m3 is even. Homework Equations Definition of even: n=2k The Attempt at a Solution Proof: Let n, m \in Z such that n-m=2k n-m=2k n=2k+m m=-2k+n...

Homework Statement If m is an odd integer and n divides m, then n is an odd integer. Homework Equations Odd integers can be written in the form m=2k+1. Since n divides m, there exists an integer p such that m=np The Attempt at a Solution We will assume that m is an odd integer and...

Abel’s Lemma, Let a_0,a_1,a_2,\cdots and b_0,b_1,b_2,\cdots be elements of a field; let s_k = a_0 + a_1 + a_2 + \cdots + a_k k= 0,1,2,… And s-1 =0. Then for any positive real integer n and for m= 0,1,2,…,n-1, \sum^n _{k=m} a_k b_k = \sum ^{n-1}_{k=m} (b_k - b_{k+1}) s_k + b_n...

1. Briefly define "reality." 2. Is a "proof" of reality possible? 3. Relate your most direct proof of reality. 4. Does, or how does, that differ from a God-proof?