# triclino

Senior Members

285

1. ## small arguments

Based on what axiom theorem or definition??
2. ## Differentials in the product rule

Definition: if y = f(x) ,then dy= f'(x)Δx ,and dx = 1.Δx. Theorem : if y = u(x)v(x) ,then dy = duv + udv. hence y+dy = vdu +udv +uv. Thus $y+dy\neq(u+du)(v+dv)$. However y+Δy = u(x+Δx)v(x+Δx) = (u+Δu)(v+Δv)
3. ## small arguments

Is that a general statement ??
4. ## small arguments

So $\neg(0<-1)$ and 1>0 are not true since the arguments are invalid?? Besides $\neg(0<-1)$ and 1>0 not only are true ,but can be logically concluded in the following way: 2>1 => 2-1>1-1 => 1>0 $\neg(1<0)\Longrightarrow 1\geq 0\Longrightarrow 1-1\geq -1\Longrightarrow 0\geq -1\Longrightarrow\neg(0<-1)$
5. ## small arguments

If we are presented with the following arguments : 1) If 1<0 ,then 1-1< -1 => 0<-1. But since $\neg(1<0)$ ,then $\neg(0<-1)$. 2) if 1>0 then 1+1>0+1 => 2>1 . And since 2>1 ,then 1>0 What true facts can we use so that we can decide whether the above arguments are valid or non valid??
6. ## Why is cosine used in the definition of the dot product

hobz ,i wonder why you don't look for an explanation in WIKIPEDIA ,the explanation there is excellent. Vector dot product is very common concept and the WEB is full of it
7. ## Is this correct?

Yes there is a typo in the book. No doubt about it
8. ## Is this correct?

The ratio of F(force) to their weight is : (mg-ma)/mg = 1-a/g
9. ## definition of sqrt(x)

If they are equivalent we must have: (1) implying (2) and (2) implying (1) . I can see that (2) implies (1) ,but how does (1) imply (2)??
10. ## definition of sqrt(x)

Can the following two definitions of sqrt(x) be considered as equivalent?? 1) if $x\geq 0$ ,then ($\sqrt x=y$) if ($y^2=x\wedge y\geq 0$) 2) if $x\geq 0$ ,then ($\sqrt x=y$) iff ($y^2=x\wedge y\geq 0$)
11. ## Definition of |x|

Yes but introducing false statements into a proof can lead us to disastrous consequences
12. ## Definition of |x|

But ,however, logic dictates us the following: If we put : ($x\geq 0$): = p (|x|=x) := r (x<0): = q (|x|= -x): = s...............then we have: ($p\Longrightarrow r$) and ($q\Longrightarrow s$) which logicaly implies: $p\wedge q\Longrightarrow r\wedge s$ But p&q means that we have : $x\geq 0$ and x<0 . IS that possible??
13. ## Definition of |x|

Can the following definition of the absolute value be considered as correct?? ($x\geq 0\Longrightarrow |x|=x$) and (x<0$\Longrightarrow |x|=-x$)
14. ## What constitutes a derivative?

Given y = f(x) ,then we define : dy = f'(x)Δx and Δx= 1.dx . Hence by definition: dy/dx = f'(x) ,where dy/dx is the ratio of the differentials dy ,dx In the case where the derivative is denoted by :$\frac{dy}{dx}$,then this is equal with the ratio of the two differentials dy/dx
15. ## uniform convergrence

You mixing up uniform continuity with uniform convergence . But in my very 1st post i ask if any body could prove uniform convergence that i could not prove. Now by producing a No and plugging it into the definition of uniform convergence to find out if the definition is satisfied or not ,i am sorry is not much of a help
16. ## empty set

O.k this a trivial proof and i am waiting for another trivial proof : Suppose x does not belong to the set ,A .But x does not belong to the empty set. Hence if x does not belong to A ,then x does not belong to the empty set . Thus : $\emptyset\subseteq A$
17. ## uniform convergrence

You mixing up ceiling function with the floor function. But according to what axiom or theorem you came to the conclusion : $N=\lceil 1/(4\epsilon^2)\rceil+1$ or $N=\lceil 1/(4\epsilon^2)\rceil$
18. ## uniform convergrence

Do you know any natural No N to be equal to $\frac{1}{4\epsilon^2}$
19. ## 3 vector product

Why you did not ask me what i meant ,but make such a fuss over minor details ?? This is a physics forum and people know what a dot product is , and very easily can understand that: A.(B.C) is really A(B.C) since the dot product is always a scalar. Now is not true that if the vectors are on the same currier or parallel then : A(B.C) =(A.B)C ???
20. ## uniform convergrence

Let : $f_{n}(x) =\frac{x}{x^2+n}$ be a sequence of functions in the Real Nos. I can prove point wise convergence to the zero function as n goes to infinity . But is there a uniform convergence ?? If there is, can anyone prove it ,please??
21. ## empty set

Perhaps i should have mentioned in my 1st post,that i have in my mind a proof. But i am interested to know if there are any other proofs. Apart of those that Wikipedia offers
22. ## empty set

That does not help in the above proof
23. ## empty set

What is the definition of the empty set??
24. ## What is a mathematical proof

In mathematics sometimes, the blatant ,when we try to prove it becomes blatantly impossibly. For example : The BOLZANO theorem
25. ## empty set

prove without using contradiction.that the empty set is the subset of every set
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