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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
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
What is the sum of 538 ans 259?
The sum of 538 and 259 is 797. The word Sum means to add.
Answered by: Emilio Aranda_EL Paso, Tx (Emilioa)
What is the difference between equal two lines and identical three lines in maths?
Three lines between two equations means that there is an identity present. This means that whatever numbers you put into the equation it will always equal the equation on the other side.
Let the triangle be ABC and its medians by AX and BY and CZ.
Therefore, since AC=AB, and OP=BD.
Therefore,
By Triangle Equiangular Property,
Triangle ABC simliar to Triangle XYZ
Therefore,
Three times the sum of the squares of sides of triangle equal to 4 times of its median.:)
Hope it helped.
What is 1000000 multiplied by 5?
The obvious answer is 5,000,000 but...
If the 1000000 is in base 2 (binary) and the five is in base 10 (decimal) then 1000000 equals 64 in base 10; 64x5 = 320 which is 101000000 in base 2.
The diagonals of a parallelogram are congruent?
No, the diagonals of a parallelogram are not normally congruent unless the parallelogram is a rectangle.
How many divisors does 9 factorial have?
9! = 9×8×7×6×5×4×3×2
Separate each of the above factors into prime and non-prime:
Prime
2
3
5
7
Non-prime
4
6
8
9
Perform a prime factorization of each of the non-prime factors:
4 = 22
6 = 3×2
8 = 23
9 = 32
Rewrite the number using these prime factorizations:
9! = 32 × 23 × 7 × (3×2) × 5 × 22 × 3 × 2
Group:
9! = 7 × 5 × 34 × 27
For any number with factors (greater than 1) of aA × bB × cC × ... × nN , it can be shown that the total number of divisors is (A+1)×(B+1)×(C+1)× ... × (N+1). This is because there are N+1 possible ways to divide out the factor n to create a unique divisor (n0 is also a factor). Using basic combinatorics, the total possible number of divisors is simply the products of all these possibilities for each prime factor.
Therefore, the number of divisors in 9! can be computed as follows:
(1+1)(1+1)(4+1)(7+1) = 160
Thus, 9 factorial has 160 divisors.
How many cents isone hundred fils?
Many countries or regions (eg USA and Eurozone) use cents as the minor denomination for their currency. Similarly, there are very many countries that use fils. Without any information about the countries in question, it is impossible to answer the question.
What is an odd number of factors?
Perfect squares have an odd number of factors.
If f is a factor of N, then so is N/f. (For example, 3 is a factor of 30, and so is 10). Thus, it seems factors come in pairs, and there should always be an even number of them. But if f = N/f, then these two factors are the same, resulting in an odd number of factors. For example, 3 is a factor of 9, and it's "mate" is also 3.
So for the numbers with an odd number of factors, there is some f where f=N/f. Multiplying both sides by f, we have f^2 = N. So this happens when N is a perfect square.
How do you derive the mean deviation about the mean of the pareto type 1 distributions?
Easy.
The mean deviation about the mean, for any distribution, MUST be 0.
How do you find the edge of a cube if the measure of the diagonal is given?
The longest diagonal in a cube is equal to the length of the edge, multiplied by the square root of 3.
What is divisible by 1234 and 6?
think of "reverse operations"...1234 times 6 would equal a # that is divisible by 1234 and 6, due to that multiplication is the reverse operation of division. or 7404
There cannot be a proof since the statement need not be true.
How simple pendulum obeys simple harmonic motion?
The acceleration of the bob is directly proportional to the displacement and towards the vertical position.If x represents the angular displacement towards the right, from the vertical. and if x', x'' represent the derivatives, then
x'' = -kx where k > 0.
This is the characteristic differential equation for SHM.
How do you find the total surface area of a cube?
Determine the area of one face (2m x 2m = 42m). Multiply by the number of faces on a cube - 6.
So, 6 x 42m = 242m total surface area.
What is the inverse of a conditional statement?
The statement formed when you negate the hypothesis and conclusion of a conditional statement.
For Example: If you had enough sleep, then you did well on the test.
The inverse will be: If you didn't have enough sleep, then you didn't do well on the test.
What would their combined surface area be?
It would be the sum of the areas of each surface separately.
What is a fallacious argument?
An argument that sometimes fools human reasoning, but is not logically valid.
Why is 2.5 km more than 2700 meters?
2.5 km is not more than 2700 meters, it is less. 2.5 km is 2500 meters.
