What is Direct proof: Definition and 17 Discussions

In mathematics and logic, a direct proof is a way of showing the
truth or falsehood of a given statement by a straightforward combination of
established facts, usually axioms, existing lemmas and theorems, without making any further assumptions. In order to directly prove a conditional statement of the form "If p, then q", it suffices to consider the situations in which the statement p is true. Logical deduction is employed to reason from assumptions to conclusion. The type of logic employed is almost invariably first-order logic, employing the quantifiers for all and there exists. Common proof rules used are modus ponens and universal instantiation.In contrast, an indirect proof may begin with certain hypothetical scenarios and then proceed to eliminate the uncertainties in each of these scenarios until an inescapable conclusion is forced. For example, instead of showing directly p ⇒ q, one proves its contrapositive ~q ⇒ ~p (one assumes ~q and shows that it leads to ~p). Since p ⇒ q and ~q ⇒ ~p are equivalent by the principle of transposition (see law of excluded middle), p ⇒ q is indirectly proved. Proof methods that are not direct include proof by contradiction, including proof by infinite descent. Direct proof methods include proof by exhaustion and proof by induction.

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  1. 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...
  2. 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...
  3. 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...
  4. 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...
  5. 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...
  6. 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...
  7. 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.
  8. 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)...
  9. 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...
  10. 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...
  11. 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 ...
  12. 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...
  13. 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...
  14. 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...
  15. 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...
  16. 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...
  17. Loren Booda

    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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