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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
How do you prove that there is a prime between n and 2n?
To be pedantic, the question should say "for all n >= 2". A detailed proof is given here: http://mathforum.org/library/drmath/view/51527.html The proof is quite long, but it only uses properties of logarithms, exponents, and the binomial theorem, so if you know about these and have enough mental stamina, you can probably make sense of it.
Which theorem is used to prove the AAS triangle congruence postulate theorem?
The first thing you prove about congruent triangles are triangles that have same side lines (SSS) is congruent. (some people DEFINE congruent that way).
You just need to show AAS is equivalent or implies SSS and you are done.
That's the first theorem I thought of, don't know if it works though, not a geometry major.
The exponential function, logarithms or trigonometric functions are functions whereas a complex variable is an element of the complex field.
Each one of the functions can be defined for a complex variable.
Reverse and negation of an if-then statement?
The reverse and negation of an if-then statement is as follows:
if (...) then statement;
reversed becomes
if (not (...)) then statement;
Let the triangle be ABC and suppose the median AD is also an altitude.AD is a median, therefore BD = CD
AD is an altitude, therefore angle ADB = angle ADC = 90 degrees
Then, in triangles ABD and ACD,
AD is common,
angle ADB = angle ADC
and BD = CD
Therefore the two triangles are congruent (SAS).
And therefore AB = AC, that is, the triangle is isosceles.
In how many different ways can the letters of the word coniohser be arranged?
Coniohser has 9 letters. So the solution is permutation of nine that means 1*2*3*4*5*6*7*8*9 =362880
Does every statement have a counterexample?
No. Not if it is a true statement. Identities and tautologies cannot have a counterexample.
What are other differences of postulate and theorem?
The phrase "other differences" implies that you already have some differences in mind. However, you have not bothered to share that information. Consequently, there is no way for me to know if a difference that I mention is one that you already know or if it is another one.
How do you prove the formula for the volume of a square pyramid?
You really can't "prove" the formula. You use it. You first square the base 'b'. Then, you multiply that number by the height 'h'. Then, you divide the product of the base squared and height by 3. Boom! You get your answer. In my school, we get a formula sheet with all the formulas we will need to use. If you didn't understand the description above, here is the formula for a square pyramid:1/3b2 h.
Hope this helped!
The question cannot be answered without more information about the points e, f and g.
When 8 is subtracted from two times a number the result is 10 what is the number?
8-(2x)=10
-(2x)=10-8
-(2x)=2
x=2/(-2)
x= -1
Prove AnB subset A subset AUB?
I shall answer this under the assumption that 'n' means intersection.
Recall the definitions of intersection and union:
1) x is an element of AnB if and only if x is an element of A and x is an element of B
2) x is an element of AUB if and only if x is an element of A or x is an element of B
and recall that
3) X is an (improper) subset of Y if and only if every element of X is an element of Y
Thus, if x is an element of AnB, then x is an element of A and an element of B, so it clearly is an element A (law of simplification in logic). This implies AnB is a subset of A. Now if x is an element of A, it is certainly an element of A or an element of B (law of addition in logic), and therefore x is an element of AUB.
There are other ways of answering this based on axiomatic approaches.
State and prove the Cochran's theorem?
can be written, where each Qi is a sum of squares of linear combinations of the Us. Further suppose that
where ri is the rank of Qi. Cochran's theorem states that the Qi are independent, and each Qi has a chi-squared distribution with ri degrees of freedom.[citation needed]
Here the rank of Qi should be interpreted as meaning the rank of the matrix B(i), with elements Bj,k(i), in the representation of Qi as a quadratic form:
Less formally, it is the number of linear combinations included in the sum of squares defining Qi, provided that these linear combinations are linearly independent.
ExamplesSample mean and sample varianceIf X1, ..., Xn are independent normally distributed random variables with mean μ and standard deviation σ thenis standard normal for each i. It is possible to write
(here, summation is from 1 to n, that is over the observations). To see this identity, multiply throughout by and note that
and expand to give
The third term is zero because it is equal to a constant times
and the second term has just n identical terms added together. Thus
and hence
Now the rank of Q2 is just 1 (it is the square of just one linear combination of the standard normal variables). The rank of Q1 can be shown to be n − 1, and thus the conditions for Cochran's theorem are met.
Cochran's theorem then states that Q1 and Q2 are independent, with chi-squared distributions with n − 1 and 1 degree of freedom respectively. This shows that the sample mean and sample variance are independent. This can also be shown by Basu's theorem, and in
