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Proofs
Proof means sufficient evidence to establish the truth of something. It is obtained from deductive reasoning, rather than from empirical arguments. Proof must show that a statement is true in all cases, without a single exception.
1,294 Questions
We prove it for the interval (0,1) and the proof is easily extended to any subset of real numbers. An alternative way to state this is every infinite set of points in (0,1) contains a sequence converging to a point not in the sequence.
So proof is by contradiction. We want to show that some subset (arbitrary) of (0,1) has a limit point. So let's assume this is NOT true.
Let K be the subset of (0,1) consisting of all reals in the interval, of course K is infinite, Now let's start by forming a sequence and let A0 be the first term and let A0 =K
An+1=An - lub(An) where lub means the least upper bound
1. Now we note that the lub of A1 exists because A1 is bounded by 1 above.
2. A1 is non-empty sine A0 is infinite and one point is removed at each step.
3. lub(Ai) ∈ Vi. Otherwise, it would be a limit point of Ai which is a contradiction.
4. lub(Ai+1) < lub(Ai). Since Ai+1 ⊂ Ai, ∀a∈Ai+1, a < lub(Ai )
Now let's look at two mutually exclusive possibilities:
Case 1: We know there is some k such that Ak =Ak+1
This can't be because it violates #3 since since Ak = Ak+1 = Ak - lub(Aj), lub(Ak) ∉ Aj
So let's say for all k Ak ≠Ak+1 which we assume since case 1 is not possible.
Then
Form the set S={lub(Ak)}. A is itself a subset of (0,1) bounded below by 0, so it has a greatest lower bound. Let s=glb(S). If s ∉ S, s is a limit point of S, a contradiction. If s ∈ S, s=lub(Ak) for some k. However, by (4), Ak+1 < s, a contradiction.
So we assume our subset does have a limit point and the proof is complete.
When trying to prove two triangles congruent, you can use SSS, SAS, ASA, AAS, HL, and HA patterns. However, the pattern A S S doesn't work. Instead of spelling or saying this word in class, you can refer to it as "the donkey theorem". You can look at the pattern in the two triangles and say "these two triangles are not congruent because of the donkey theorem."
You CANNOT prove triangles incongruent with 'the donkey theorem', nor can you prove them congruent. It's mostly sort of a joke, you could say, but it's never useful.
The reason is that if the two triangles ARE congruent, then of course there will be an unincluded congruent angle as well as two congruent sides.
The theorem doesn't do anything left, right, forward or backward. It's not even really a theorem. :P
Differentiation Rules and examples and explanation?
Differentiating is the act of finding the derivative of a function, thus allowing you to find out how the function changes as its input changes, such as finding the rate of change of the gradient of the function, and when differentiating with respect to time can allow you to give equations explaining the motion of various objects, depending on how you differentiate the functions. There are two main types of differentiation, ordinary differentiation and partial differentiation, the rules outlined below are for ordinary differentiation. While the rules for partial differentiation are not that dissimilar, they do not need to be known outside of university level mathematics and physics.
(using f' and g' to denote the derivative of the functions f and g of x respectively, x is a variable, o indicates a composite function, all other letters are constants, rules in bold are important)
Elementary rulesf = xn, f' = nx(n-1) elementary power rule
f = a, f' = 0 constant rule
f = ax, f' = a derivative of a linear function is a constant
(af +bg)' = af'+bg' linearity of differentiation, leading to the 3 following,
(af)' = af' constant multiple rule
(f+g)' = f'+g' sum rule
(f-g)' = f'-g' subtraction rule,
(fg)' = f'g + fg' product rule
(fog)' = (f(g))' = (f'(g))g' = (f'og)g' chain rule
f = 1/g, f' = -g'/g2 reciprocal rule
(f/g)' = (f'g-fg')/g2 quotient rule
f=sin(x), f'=cos(x)
f=cos(x), f'=-sin(x)
f=tan(x), f'=sec2(x)
f=sec(x), f'=sec(x)tan(x)
f=cosec(x) f'=-cosec(x)cot(x)
f=cot(x), f'=-cosec2(x)
f=exp(ax), f'=a*exp(ax)
f=exp(axn), f'=anx(n-1)*exp(axn)
f=ax, f'=(log(a))*ax
f=log(x), f'=1/x
f=log(xn), f'=nx(n-1)/xn
f=xx, f'=xx(1+log(x))
these are just the simple rules of differentiation for various functions, there are a LOT more, but they are generally only of use at university levels.
The sum is 22 times the sum of the three digits.
Yes they can be so friendly, in fact, that you could have one as your pet!!! HOW COOL WOULD THAT BE!!!! For more fox info email me on jimmyboi14@hotmail.co.uk
12 and 18
Who is the father of vedic maths?
Jagadguru Shankaracharya Shri Bharati Krishna Tirthaji Maharaja is 2the father of vedic maths
Is every statement a theorem why?
There are many kinds of statement that are not theorems:
A statement can be an axiom, that is, something that is assumed to be true without proof. It is usually self-evident, but like Euclid's parallel postulate, need not be.
A statement need not be true in all circumstances - for example, A*B = B*A (commutativity) is not necessarily true for matrix multiplication.
A statement can be false.
A statement can be self-contradictory for example, "This statement is false".
If 0 divided by 0 is equal to zero then prove that 1 equal to 2?
(2 - 1) * 0 = 0
Thus 2 - 1 = 0/0 = 0
and therefore 2 = 1
You start with one piece of paper. With each cut you increase the number of pieces by one - whether or not you cut in half. So, at the end of the process you have as many pieces as the number of cuts you made and that, in turn, depends on the rate at which you cut the paper and the time available - neither of which are given in the question.
How are the proofs of the fundamental theorem of algebra?
look in google if not there, look in wikipedia.
fundamental theorem of algebra and their proofs
