triclino

Based on what axiom theorem or definition??

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 [math]y+dy\neq(u+du)(v+dv)[/math]. However y+Δy = u(x+Δx)v(x+Δx) = (u+Δu)(v+Δv)

Is that a general statement ??

So [math]\neg(0<-1)[/math] and 1>0 are not true since the arguments are invalid?? Besides [math]\neg(0<-1)[/math] and 1>0 not only are true ,but can be logically concluded in the following way: 2>1 => 2-1>1-1 => 1>0 [math]\neg(1<0)\Longrightarrow 1\geq 0\Longrightarrow 1-1\geq -1\Longrightarrow 0\geq -1\Longrightarrow\neg(0<-1)[/math]

If we are presented with the following arguments : 1) If 1<0 ,then 1-1< -1 => 0<-1. But since [math]\neg(1<0)[/math] ,then [math]\neg(0<-1)[/math]. 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??

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

Yes there is a typo in the book. No doubt about it

The ratio of F(force) to their weight is : (mg-ma)/mg = 1-a/g

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)??

Can the following two definitions of sqrt(x) be considered as equivalent?? 1) if [math]x\geq 0[/math] ,then ([math]\sqrt x=y[/math]) if ([math]y^2=x\wedge y\geq 0[/math]) 2) if [math]x\geq 0[/math] ,then ([math]\sqrt x=y[/math]) iff ([math]y^2=x\wedge y\geq 0[/math])

Yes but introducing false statements into a proof can lead us to disastrous consequences

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

Can the following definition of the absolute value be considered as correct?? ([math]x\geq 0\Longrightarrow |x|=x[/math]) and (x<0[math]\Longrightarrow |x|=-x[/math])

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 :[math]\frac{dy}{dx}[/math],then this is equal with the ratio of the two differentials dy/dx

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

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 : [math]\emptyset\subseteq A[/math]

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

Do you know any natural No N to be equal to [math] \frac{1}{4\epsilon^2}[/math]

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 ???

Let : [math]f_{n}(x) =\frac{x}{x^2+n}[/math] 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??

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

That does not help in the above proof

What is the definition of the empty set??

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

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