# What is implication: Definition and 73 Discussions

In formal concept analysis (FCA) implications relate sets of properties (or, synonymously, of attributes). An implication A→B holds in a given domain when every object having all attributes in A also has all attributes in B. Such implications characterize the concept hierarchy in an intuitive manner. Moreover, they are "well-behaved" with respect to algorithms. The knowledge acquisition method called attribute exploration uses implications.

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1. ### Help Needed Proving Implication for Linear Functional on Banach Space

Dear everybody, I am having some trouble proving the implication (or the forward direction.) Here is my work: Suppose that we have an arbitrary linear functional ##l## on a Banach Space ##B## is continuous. Since ##l## is continuous linear functional on B, in other words, we want show that...
2. ### I Question about implication from scalar product

Hi, Let's say we have the Gram-Schmidt Vectors ##b_i^*## and let's say ##d_n^*,...,d_1^*## is the Gram-Schmidt Version of the dual lattice Vectors of ##d_n,...,d_1##. Let further be ##b_1^* = b_1## and ##d_1^*## the projection of ##d_1## on the ##span(d_2,...,d_n)^{\bot} = span(b_1)##. We have...
3. ### B Vacuously true statements and why false implies truth

We say that an implication p --> q is vaccuously true if p is false. Since now it's impossible to have p true and q false. That is we can't check anymore whether the contrary, p being true and q being false,can be.Since p being true is non-existent. So we take the implication as true. For eg...
4. ### I Logical implication vs physical causality

There is something I don't understand that I want to ask quantum physics experts here: Suppose the happening of event X results logically speaking in the happening of event A. So we could for instance have the following logical implication ##X.happens \rightarrow A.happens##. If this is...
5. ### B Implication of Goedel's theorems on physics

I am a graduated mathematician. Do you know that Kurt Godel proofed that it is not possible to proof that math is not noncontradiction? Physicist and a lot of other sciences are based od math. So, how can you proof that whole physics' is noncontradiction?You just believe in it as someone in God?
6. ### I Translate compound proposition p → q (implication) to p↓q question

I hope someone can help me or point me in the right direction. I am reading Discrete Mathematics with its Applications by Rosen. I am trying to self learn discrete math. I am actually able to do most questions but I have a question about a solution (not the question itself.) The question is...
7. ### I Why are the inverse and converse of an implication not equivalent?

If we have the statement: "If we prepare, we'll win the war", then according the rules of the truth table for this implication, this statement is only false if we prepared and still lost the war. This is what I'm having trouble with about implication. I understand that the only way to falsify...
8. ### I Direction of logical implication in bijectively related sets

I have a hypothesis of which I wonder if it's sound. Perhaps you guys can advise me: Suppose ##x_n\Rightarrow a_n## (logical implication) for some set X and set A. I think we have to assume a bijection. Then, if ##a_m = False##, ##x_m## should be ##False##, right? So, in case of a bijection...
9. ### I Implication, Boolean Expression and Venn Diagrams

Hello, I’m having difficulties understanding logical relations “A implies B” and “A if and only if” using Boolean expressions and Venn diagrams, there is something where I’m wrong, but I could not find it out. Please, be benevolent and tell me where I’m wrong. Thanks Note : Obviously I’m...
10. ### I Implication of length contraction

I had posted a similar question on another forum but didn't get much of a discussion. I'm interested to know what people here think. So consider a spaceship midway between stars A and B and initially at rest in the reference frame of the stars. The ship then accelerates away from A to some...
11. ### I Understanding Implication Problems: Examples & Explanations

I just saw this proof... And, I don't understand why this is true. How am I supposed to think about problems like this one? Edit: Here's another one: The only steps here I understand are 1 to 5. I don't know why 6 and 7 are true.
12. ### MHB Proposition 2.2.9: Understanding the Implication of Lemma 1.1.7 (iv) for Peter

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 2: Differentiation ... ... I need help with an aspect of the proof of Proposition 2.2.9 ... ... Duistermaat and Kolk's Proposition 2.2.9 and its proof read as...
13. ### MHB Natural deduction (negated implication)

The proposition ¬(P→Q) is equivalent to ¬P^Q Does someone maybe have an idea how you can prove (directly) ¬P^Q from ¬(P→Q) by means of natural deduction? I do not manage it. Thanks in advance!
14. ### I Correct way to understand logical implication?

If an object type ◌ exists and within the set of these objects exist some which have both well defined properties A and B, and some which only have the property A. There is at least one ◌ which does not have both properties. Thus say we create the set of such objects such that A or B (inclusive...
15. ### MHB Logical Implication: Evaluating Truth of Claims

Hello all, I am trying to find if the following two claims are true or false: 1) If $\alpha \models \left ( \beta \rightarrow \gamma \right )$ then $\alpha ,\beta \models \gamma$ 2) If $\alpha \models \left ( \beta \rightarrow \gamma \right )$ then $\alpha \vee \beta \models \gamma$...
16. ### I Implication of eigenstates that are not normalizable

If the spectrum of a hermitian operator is continuous, the eigenfunctions are not normalizable. I have been told that these eigenfunctions do not represent possible physical states, but what exactly does that mean? Is there a good interpretation of the physicality of these eigenstates? If a...
17. ### Unpacking the Implication of a Formula: n=2

Edited by moderator In the attached file it says that the formula for when n=2 implies the next one but I don't understand why it suddenly became (s-t0)^2 inside the integral instead of (s-t0) in the next one and keeps this pattern, it doesn't seem to be implied anywhere for me.
18. ### How Do Bilinears Connect to Killing Vectors and Differential Forms?

What does it mean that a Killing vector and a total differential of a certain theory are related to bilinears? In other words, why would bilinears (e.g. of the forms ##<\gamma_0\epsilon, \gamma_5\gamma_{\mu}\epsilon>## and ##<\gamma_0\epsilon, \gamma_{\mu}\epsilon>## tell us anything about...
19. ### Anomaly set theoretic equivalent to material implication

OK, this is embarrassing, but I never looked carefully at this elementary point. We say that if p implies q P is the set of all things for which p is true Q is the set of all things for which q is true then Q ⊆ P. Also that the set of all things for which p&q is true equals P∩Q But p & q...
20. ### MHB Logic equivalence and implication, question

Hey guys I am having a bit of a difficult time with this question, if some one could help me out it would be appreciated, thanks. Consider the following argument. "If the weather is fine, and the train is early, then the dog will sit on the tuckerbox. The train will be early, (but) also there...
21. ### Second order curiosities: MPP, implication

In setting up a syntax for a first order theory, one usually includes Modus Ponens as a metarule. However, couldn't MP just be rewritten as a second-order sentence, thereby making all supposedly first-order theories de facto second-order ones? While on the subject of second order theories, I...
22. ### "Members of a frame are axially inextensible" implication?

In analysis of frames, we say that "The members of the frames are axially inextensible". In context of frames with vertical legs, what I understand is that the length of the chord joining the member ends in the displaced condition would be same in the axial direction. However, what does it imply...
23. ### MHB Why does the implication hold?

Hi! (Smile) I am looking at the proof of the following sentence: For each natural number $n$ it holds that $n \notin n$. Proof : We define the set $X=\{ n \in \omega: n \notin n\}$. It suffices to show that $X$ is an inductive set, because then $X=\omega$. Obviously $\varnothing \in X$. We...
24. ### Implication of Hardy's ontological excess baggage theorem

Given Hardy's "excess baggage" theorem showing that the size of the ontic state space must scale exponentially with the number of systems, does this necessarily pose any "threat" to the ontic approach to QM? Leifer writes: Is the quantum state real? A review of ψ-ontology theorems...
25. ### Necessity Condition in Implication ?

P -> Q P Q P->Q T T T T F F F T T F F T I understand the condition "P is sufficient for Q". But I'm not getting the meaning of why "Q is necessary for P". What does this signifies, Please explain ..
26. ### MHB Why is the implication obvious?

Hello! (Smile) I am looking at the proof of the follwing sentence: Let $a,b \neq 0$. The common multiples of $a,b$ are the same as the multiples of $[a,b]$, where $[a,b]$ is the least common multiple of $a \text{ and } b$. Let $a \mid m, b \mid m$ m=q \cdot [a,b]+r , \ 0 \leq r < [a,b]...
27. ### Inflation and it's implication

HI,... This is my first post here on this forum ... I wonder if anyone here can help me to clarify some concepts to me in the paper of alan guth 2007 named "Eternal inflation and its implications" and this is my question here ... he says in the abstract the following :"Although inflation is...
28. ### Set Theory: Why use implication?

Homework Statement I hope this does not violate copyright or anything but this problem originated from an assignment from Introduction to Mathematical Thinking in Coursera. I could not post there because the class ended and the discussion board there is dead. Let C be the set of all cars, let...
29. ### MHB Material Implication: True Premise, False Result

I'll be linking to this post when the question arises why an implication with a false premise is true.
30. ### Sun-synchronous orbit: Implication on the orbit's inclination

Homework Statement A satellite is launched into a circular sun-synchronous orbit at a height of 900km above Earth's surface. What is the implication on the orbit's inclination (in deg) and on the change of the position of the right ascension of the ascending node per day.Homework Equations The...
31. ### Truth table, implication and equivalence

Hello, I have some questions about the truth tables for impliocation and equivalence. for implication we have: p | q | p=> q T | T | T T | F | F F | T | T F | F | T Here I do not understand the last two lines, how can we say that p implies q when...
32. ### Implication of a set of zeros with positive measure

I have a non-zero measured subset X\subseteq\mathbb{R}^{n} on which \sum_{i=1}^{n}\psi_{i}x_{i}=0 for all x=(x_{1},\ldots,x_{n}) in X. How can I show that \psi_{i}=0 for i=1,\ldots,n?
33. ### The output would be Logical Implications: A => B and B => A.

A: (p => ~q) ^ (p v q) B: ~p v q does A => B (A implies B) ? does B => A ( B implies A) ? I did the truth tables for each: A => B: http://www4c.wolframalpha.com/input/?i=%28%28p+%3D%3E+NOT+q%29+AND+%28p+OR+q%29%29+%3D%3E+%28NOT+p+OR+q%29 B => A...
34. ### Validity of Implication Arguments in Movie Reviews

Homework Statement Is the argument below valid? If it's valid, write down the argument symbolically. If a movie is not worth seeing, then it was not made in England. A movie is worth seeing only if critic Ivor Smallbrain reviews it. The movie The Good, The Bad and The Mathematician was...
35. ### Lorentz Contraction ?s and Implication

Unfortunately, my knowledge on this is limited to wiki which I trust is relatively correct but I would like to clear up some ambiguity. (I don't mind technical but please back it up with a general explanation since I am completely new to this. The general explanations I value more.) Wiki...
36. ### Logic - clarification needed about implication

If P→Q, and P is false but Q is true, then why is P→Q true? To me, it seems as though we shouldn't be able to do proceed because there isn't enough information. Same goes when P and Q are both false, how does that suggest P→Q is true?
37. ### A question about implication in logic

According to the truth tables in my computer architecture text, P → Q is false if P is true and Q false; and true otherwise. I cannot understand why it is "true" otherwise. For example, if P and Q are both true, P → Q is also true, but this makes no sense to me. Perhaps Q is true for...
38. ### Help understand FTL causality implication

First of all, I am fairly new to relativity, but not clueless. I am not saying that FTL is possible. I am not denying relativity principles. I am stating that FTL may be plausible. Relativity gives flexibility to how you can synchronize clocks and that does not affect outcomes of most...
39. ### MHB On the Implication of Addition Axioms

Hello everyone! I was trying to prove the propositions that follow the addition axioms as a revision, I got a different proof for the following proposition: If $x+y=x+z$ then $y=z$ My proof was the following: $x+y=x+z$, $(-x)+x+y=(-x)+x+z$, $0+y=0+z$, $y=z$ Rudin however, in his book...
40. ### An Implication of a limit rule

Homework Statement If lim (x->∞) [ln(x^(1/x))]=0 and lim (x->∞) x^(1/x)=1, then does this => lim (x->∞) [ln(x^(1/x))]= ln(lim(x->∞) [(x^(1/x))]) = ln(1)?? Homework Equations lim (x->∞) [ln(x^(1/x))]=0 and lim (x->∞) x^(1/x)=1 The Attempt at a Solution lim (x->∞) [ln(x^(1/x))]=...
41. ### Hypothetical implication of Lorentz contraction

Would it be possible for a truck, measuring 5ft tall when stationary, to pass under a 4ft barrier by accelerating towards the speed of light? If so what would a spectator see if standing next to the barrier, would the spectator see the truck shrink? What would you see of the barrier from the...
42. ### Rules of Implication: Is [(P implies A) etc] True?

Suppose I know that [(P implies ~A) and (P implies B) and (P implies C)] is impossible. Does this means that the following statement is true: [(P implies A) and (P implies B) and (P implies C)]? Any help is greatly appreciated!
43. ### MHB The little theorem and its implication of coprimality.

Hi guys. I'm rather new to number theory, and as part of an assignment I have been learning about various different primality tests. One of these tests is the Lucas primality test. As part of the reasoning behind the test, wikipedia states: "If [$a^{n-1} \equiv 1 \textrm{ (mod }n\textrm{)}$]...
44. ### Why/How does the definition of implication in mathematics work?

I understand that we just have to fill the last two raws in the truth table with any value, and that we randomly chose True, and that the value True makes matters easier sometimes (I don't know an example of that, but I read that somewhere). But the question is, since mathematics is tied to...
45. ### Implication (Discrete math logic)

The truth table for implication looks like this p|q| p -> q ------------ T|T | T T|F | F F|T | T <----I'm trying to make sense of this one. My prof warned us that its strange. F|F | T I that implication means: "If p, then q" "q is necessary for p" "p is sufficient for q" "p, only...
46. ### Question about if and then statements. IE implication statements.

Question about "if and then" statements. IE implication statements. Homework Statement When something is for example asking for: if |x-3|<δ, prove that |x+3| <δ + k (where k is a constant) are they supposing it's true? Like are they giving you a hypothesis? How do implication...
47. ### The Negation of an Implication Statement?

Hello, So someone just asked me for assistance on a proof, and while I'm fairly certain you can't do what he did, I am not completely sure on the reasons. To state it as formal logic, If you have proposition A: P \rightarrow Q And let's call proposition B \neg (P \rightarrow Q) If you were to...
48. ### Electric field boundary equation implication at air/earth interface

The greatest problem of thermoelectrics is the need to maintain very low thermal conductivity. How is it possible that pyroelectrics do not have this limitation and do not need temperature differences to produce electricity from heat?If we will heat all pyroelectric body uniformly it will still...
49. ### Proof by induction (Implication)

Homework Statement The problem statement is as follows: Let T be a ordered list(sequence) with N entrys. Prove by induction that, j-i< 2^{n} \Rightarrow A(i,j) \leq n for all integers i, j, n where 1 \leq i \leq j \leq N and n \geq 0 A(i, j) is the number of steps taken by a certain...
50. ### Logical Implication - If p, then q

If p, then q. Suppose p is false but q is true. Why is it that the implication "If p, then q" is still true? For example, If x=2, then x + 3 = 5. Suppose x is NOT 2 (i.e. p is false), but x+3=5 (q is still true). Why is the implication "If x=2, then x + 3 = 5" still true? Is the...