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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
In Einstein's theory of gravitation, which is also referred to as General Relativity, space-time is warped (i.e., curved) by the presence of mass. It is not meaningful to speak of folding time alone, but rather one speaks of bending space-time. To make a significant bend in space-time, a very massive object (such as a star) or a very dense object (such as a black hole) is required. In 1915, Einstein succeeded in writing down an equation that describes how much space-time is curved in the presence of mass. That equation is a critical part of his General Theory of Relativity. It is not a simple equation to understand, since it requires first some understanding of tensors and Riemannian geometry. You will have to study math and physics for several years in order to understand it.
say peste ang tatay mo and then you'll find the answer if you do.
Is this biconditional statement true A number is divisible by 6 if and only if it is divisible by 3?
How can the following definition be written correctly as a biconditional statement?
An odd integer is an integer that is not divisible by two.
(A+ answer) An integer is odd if and only if it is not divisible by two
Why cant a quadrilateral have exactly 3 right angles?
A quadrilateral can only have either 2 right angles or four. As you may (or may not) know, the angles in a quadrilateral must add up to 360 degrees. If you have 3 right angles that adds up to 270 degrees, so the last side must also be 90 degrees.
What are the different kinds of reasoning in geometry?
*analogy- comparison if two things alike in some aspects or they will be semble one another in other aspect. *intuiton - mental ability same as guessing. *inductive - forming of general statement from particular specific case. *deductive - making conclusion from general to particular stances.
What are the square roots up to 225?
You can get these quickly with Excel, or a similar spreadsheet program. In column A, write 1, 2, 3, ... downward. In Excel, after typing 1 and 2, you can select the two cells and drag the corner (a little black square) downward) to quickly get additional numbers.Next, in cell B1, type:
=sqrt(A1)
Then copy this cell downward.
What are the end products in food making process?
The end products are substances which the consumer - human or animal - may safely eat (this does not necessarily imply they are nutritious, just that they're not immediately harmful in normal circumstances).
Basic Proportionality Theorem says: If a line is drawn parallel to one side of the triangle to intersect the other two sides at distinct points .Then the other two sides are divided in the same ratio.
PROOF ( to follow this proof, just draw the triangles and segments)
Draw triangle PQR and construct line L parallel to segment QR.
Line L intersects segment PQ and segment PR at S and T respectively.
We want to show that length of PS/ length of QS is equal to length PT/ length of PR since that is what the BPT says.
Construct segments SR and QT.
Look at triangles PTS and QTS and note they have the same height which implies that
the area of triangle PTS/ area of triangle QTS is equal to PS/ SQ.
By the same reasoning, the areas of triangle SPT/ triangle SRT is equal to PT/TR.
Triangles QTS and SRT both have the same height and both have ST as a base segment so they have the same area.
So the ratio of the area of triangle PTS to the area of triangle QTS is equal to the ratios of the area of triangles SPT/SRT.
So the ratio of PS/SQ is equal to PT/TR
Since line L which is parallel to segment QR divides segment PQ and segment PR in the same ratio we have proved the BPT.
a googolplex is 1 followed by a google zeros
a google is 1 followed by a hundred zeros
the corporation google headquarters are known as the Googleplex
Prove that for all positive integers n 10 power n equals 1 in modulo 9?
Mmm, this is a really good quality question! I mean, once you've thought about it for a second, it seems so obvious. 10^3-1 = 999, 10^5-1 = 99999, and in general anything of the form of 10^n - 1 will be divisible by 9 (because the number is made up of nothing but 9's!)
This means that all numbers of the form 10^n (where n is an integer) will be equal to 1 mod 9 (i.e., they will equal 9k + 1, where k is an integer.)
However, such an argument is far from constituting a mathematical proof. So how are we to go about proving this? Well, can we manipulate the equation to show that
10^n - 1 = 9k (k,n are integers) ???????
While this may be possible, such manipulations are not easy. Another option is to try to find a proof by contradiction: lets say that 10^n is not congruent to 1 mod 9 for some value of n. Maybe it is congruent to 2 mod 9. Does this lead to a contradiction? If it does, we still aren't done the proof: we would have to show that 10^n being congruent to 3 mod 9, 4 mod 9, etc. all lead to contradictions. Once again, this is possiblly possible, but good maths is in general about being lazy, so rather than all that work, we'd rather try and find something simpler.
And then we finally run upon the idea of mathematical induction. If you don't know what that is, see here: http://en.wikipedia.org/wiki/Mathematical_induction
Mathematical induction is an exceptionally powerful tool one can use, in particular when trying to find proofs related to the integers. We basically assume that what we are trying to prove is correct, and then see if we can find a proof that this assumption necessarily leads to the statement being correct for the next greatest number. So, we assume a formula works for any integer n, and then try and show it works for n + 1. If we are successful, then all we need to show is that the formula works for n = 1, and we will have proven it for 1,2,3,4,5,6... indeed for every integer.
So can we do that here? Well, lets firstly assume that 10^n = 1 mod 9. This means that 10^n = 9k + 1 (k element of z). Even though we want to prove that, lets just for the time being assume it's true.
Now we need to prove that 10^(n+1) = 9s + 1 (where n and s are integers).
10^(n+1) = 10^n * 10 = 9s + 1
therefore (9s + 1)/10 = 10^n
and we have assumed that 10^n = 9k + 1
therefore (9s + 1)/10 = 9k + 1
9s + 1 = 90k + 10 = 90k + 9 + 1
9s + 1 = 9 (10k + 1) + 1
as 10k + 1 is an integer, s is an integer, so therefore
10^(n+1) = 9s + 1 (where n and s are integers)
so therefore, if 10^n is congruent 1mod9, 10^(n+1) is also congruent 1mod9
All we have left to do is show that the formula works for n = 1
10^1 = 10 = 9*1 + 1. It is obvious that 10 is congruent 1mod9. And that's the proof. End it with a statement like: therefore 10^n is congruent to 1 mod 9 for all n greater than or equal to 1, by mathematical induction
Hope that helps someone out there
Who is mathematician Robert LLewellyn?
Robert Llewellyn was born in Devon in 1909. He read Mathematics at Pembroke College, Cambridge, and went to teach at Westminster School in 1932. While still teaching there, he studied theology in Ely and was ordained in 1936 at St Paul’s Cathedral. He continued to teach and spent his off-duty time meditating in the St Faith's Chapel of Westminster Abbey.
In 1939 Llewelyn went to join a missionary community in India for a year. However, the outbreak of World War II made it impossible for him to return to England. He was asked to set up a school (on the site of a derelict school) for the children of British officials who could not be returned to England to continue their education.
What does it mean if the standard deviation is greater than the mean?
The standard deviation and the arithmetic mean measure two different characteristics of a set of data. The standard deviation measures how spread out the data is, whereas the arithmetic mean measures where the data is centered. Because of this, there is no particular relation that must be satisfied because the standard deviation is greater than the mean.
Actually, there IS a relationship between the mean and standard deviation. A high (large) standard deviation indicates a wide range of scores = a great deal of variance. Generally speaking, the greater the range of scores, the less representative the mean becomes (if we are using "mean" to indicate "normal"). For example, consider the following example:
10 students are given a test that is worth 100 points. Only 1 student gets a 100, 2 students receive a zero, and the remaining 7 students get a score of 50.
(Arithmetic mean) = 100 + 0(2) + 7(50) = 100 + 0 + 350 = 450/10 students
SCORE = 45
In statistics, the median refers to the value at the 50% percentile. That means that half of the scores fall below the median & the other half are above the median. Using the example above, the scores are: 0, 0, 50, 50, (50, 50), 50, 50, 50, 100. The median is the score that has the same number of occurrences above it and below it. For an odd number of scores, there is exactly one in the middle, and that would be the median. Using this example, we have an even number of scores, so the "middle 2" scores are averaged for the median value. These "middle" scores are bracketed by parenthesis in the list, and in this case are both equal to 50 (which average to 50, so the median is 50). In this case, the standard deviation of these scores is 26.9, which indicates a fairly wide "spread" of the numbers. For a "normal" distribution, most of the scores should center around the same value (in this case 50, which is also known as the "mode" - or the score that occurs most frequently) & as you move towards the extremes (very high or very low values), there should be fewer scores.
What are the two branches of Trigonometry?
The two branches of trigonometry are plane trigonometry, which deals with figures lying wholly in a single plane, and spherical trigonometry, which deals with triangles that are sections of the surface of a sphere.
How do you prove the isosceles triangle theorem?
The isosceles triangle theorem states: If two sides of a triangle are congruent, then the angles opposite to them are congruent Here is the proof: Draw triangle ABC with side AB congruent to side BC so the triangle is isosceles. Want to prove angle BAC is congruent to angle BCA Now draw an angle bisector of angle ABC that inersects side AC at a point P. ABP is congruent to CPB because ray BP is a bisector of angle ABC Now we know side BP is congruent to side BP. So we have side AB congruent to BC and side BP congruent to BP and the angles between them are ABP and CBP and those are congruent as well so we use SAS (side angle side) Now angle BAC and BCA are corresponding angles of congruent triangles to they are congruent and we are done! QED. Another proof: The area of a triangle is equal to 1/2*a*b*sin(C), where a and b are lengths of adjacent sides, and C is the angle between the two sides. Suppose we have a triangle ABC, where the lengths of the sides AB and AC are equal. Then the area of ABC = 1/2*AB*BC*sin(B) = 1/2*AC*CB*sin(C). Canceling, we have sin(B) = sin(C). Since the angles of a triangle sum to 180 degrees, B and C are both acute. Therefore, angle B is congruent to angle C. Altering the proof slightly gives us the converse to the above theorem, namely that if a triangle has two congruent angles, then the sides opposite to them are congruent as well.
Is the square root of 6 irrational?
Answer: The square root of 6 is irrational. Reason: Just try to convince it otherwise, you will see their is no way to deal with it since it becomes angry and irrational! But seriously, you can't write it as a fraction of the from p/q with p and q being integers so yes it is irrational. The proof would be easy by contradiction.
What are facts about the number 46?
In chemistry, 46 is the atomic number of Palladium.
In human biology, there are 46 human chromosomes. There are 22 pairs of autosomes and one pair of sex chromosomes.
- In geometry, the name of a 46 sided shape is called a tetracontakaihexagon.
What is the largest number of vertices in a graph with 35 edges if all vertices are?
36 vertices if all of them are or order two except one at each end.
Give a proof that the square root of 7 is an irrational number?
Proof by contradiction: suppose that root 7 (I'll write sqrt(7)) is a rational number, then we can write sqrt(7)=a/b where a and b are integers in their lowest form (ie they are fully cancelled). Then square both sides, you get 7=(a^2)/(b^2) rearranging gives (a^2)=7(b^2). Now consider the prime factors of a and b. Their squares have an even number of prime factors (eg. every prime factor of a is there twice in a squared). So a^2 and b^2 have an even number of prime factors. But 7(b^2) then has an odd number of prime factors. But a^2 can't have an odd and an even number of prime factors by unique factorisation. Contradiction X So root 7 is irrational.
State and prove Lagrange's theorem?
THEOREM:
The order of a subgroup H of group G divides the order of G.
First we need to define the order of a group or subgroup
Definition:
If G is a finite group (or subgroup) then the order of G is the number of elements of G.
Lagrange's Theorem simply states that the number of elements in any subgroup of a finite group must divide evenly into the number of elements in the group. Note that the {A, B} subgroup of the Atayun-HOOT! group has 2 elements while the Atayun-HOOT! group has 4 members. Also we can recall that the subgroups of S3, the permutation group on 3 objects, that we found cosets of in the previous chapter had either 2 or 3 elements -- 2 and 3 divide evenly into 6.
A consequence of Lagrange's Theorem would be, for example, that a group with 45 elements couldn't have a subgroup of 8 elements since 8 does not divide 45. It could have subgroups with 3, 5, 9, or 15 elements since these numbers are all divisors of 45.
Now that we know what Lagrange's Theorem says let's prove it. We'll prove it by extablishing that the cosets of a subgroup are
- disjoint -- different cosets have no member in common, and
- each have the same number of members as the subgroup.
Lemma:
If H is a finite subgroup of a group G and H contains n elements then any right coset of H contains n elements.
Proof:
For any element x of G, Hx = {h • x | h is in H} defines a right coset of H. By the cancellation law each h in H will give a different product when multiplied on the left onto x. Thus each element of H will create a corresponding unique element of Hx. Thus Hx will have the same number of elements as H.
Lemma:
Two right cosets of a subgroup H of a group G are either identical or disjoint.
Proof:
Suppose Hx and Hy have an element in common. Then for some elements h1 and h2 of H
h1 • x = h2 • y
This implies that x = h1-1 • h2 • y. Since H is closed this means there is some element h3 (which equals h1-1 • h2) of H such that x = h3 • y. This means that every element of Hx can be written as an element of Hyby the correspondence
h • x = (h • h3) • y
for every h in H. We have shown that if Hx and Hy have a single element in common then every element of Hx is in Hy. By a symmetrical argument it follows that every element of Hy is in Hx and therefore the "two" cosets must be the same coset.
Since every element g of G is in some coset (namely it's in Hg since e, the identity element is in H) the elements of G can be distributed among H and its right cosets without duplication. If k is the number of right cosets and n is the number of elements in each coset then |G| = kn.
Alternate Proof:
In the last chapter we showed that a • b-1 being an element of H was equivalent to a and b being in the same right coset of H. We can use this Idea establish Lagrange's Theorem.
Define a relation on G with a ~ b if and only if a • b-1 is in H. Lemma: The relation a ~ b is an equivalence relation.
Proof:
We need to establish the three properties of an equivalence relation -- reflexive, symmetrical and transitive.
(1) Reflexive:
Since a • a-1 = e and e is in H it follows that for any a in G
a ~ a
(2) Symmetrical:
If a ~ b then a • b-1 is in H. Then the inverse of a • b-1 is in H. But the inverse of a • b-1 is b • a-1 so
b ~ a
(3) Transitive:
If a ~ b and b ~ c then both a • b-1 and b • c-1 are in H. Therefore their product (a • b-1) • (b • c-1) is in H. But the product is simply a • c-1. Thus
a ~ c
And we have shown that the relation is an equivalence relation.
It remains to show that the (disjoint) equivalence classes each have as many elements as H.
Lemma:
The number of elements in each equivalence class is the same as the number of elements in H.
Proof:
For any a in G the elements of the equivalence class containing a are exactly the solutions of the equation
a • x-1 = h
Where h is any element of H. By the cancellation law each member h of H will give a different solution. Thus the equivalence classes have the same number of elements as H.
One of the imediate results of Lagrange's Theorem is that a group with a prime number of members has no nontrivial subgroups. (why?)
Definition:
if H is a subgroup of G then the number of left cosets of H is called the index of H in G and is symbolized by (G:H). From our development of Lagrange's theorem we know that
|G| = |H| (G:H)
What are the kinds of trigonometry?
The main kinds are plane trigonometry and solid trigonometry. The latter will include trigonometry in hyper-spaces.
