Direct proof Definition and 18 Threads

  1. L

    B Proving the area formula for a rectangle for all positive real numbers

    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.
  2. C

    Simple Induction Direct Proofs regarding Induction

    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...
  3. B

    Mathematical Analysis Proof: |x-y|<= |x|+|y|

    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...
  4. J

    MHB Proving an Integral with a Direct Proof & Epsilon Argument

    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...
  5. M

    I For direct proof, how do you choose M for bounded sequence?

    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...
  6. M

    MHB How to prove an inequality with a direct proof?

    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 be appreciated...
  7. J

    Analyzing Logical Arguments: Not A, B or Not C, B→ (A and D), E→(C)

    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...
  8. H

    Proving the Theorem: p!/[(p-i)! * i] = 1/p for Prime Number p and Integer i

    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.
  9. J

    (Symbolic Logic) Proving P v P = P (Idempotency) using a direct proof

    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)...
  10. U

    Trying to do a non-rigorous direct proof

    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...
  11. W

    Direct proof by using if then technique

    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...
  12. H

    Proving the Oddness of 3n-2 Using Direct Proof

    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 ...
  13. M

    Proof: If n + 5 is odd, then 3n + 2 is even | Simple Direct Proof

    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...
  14. N

    Why are implications used instead of equivalent expressions in a direct proof?

    "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...
  15. C

    Direct proof using definiton of even

    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...
  16. S

    Direct Proof With Odd Integers

    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...
  17. Integral

    Abel's Lemma: Direct and Induction Proof

    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...
  18. Loren Booda

    What Is the Most Direct Proof of Reality?

    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?
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