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Hi
What is this formula called 2^n which is used to find number of distinct combinations? Does this have anything to with permutation? Please help me with this. Thanks.
HI
Please your answer simple. Thank you
Order Properties ( of real numbers ):
The properties satisfied by the relation < ( less than ) in the field R of real numbers. The basic properties are:
(1): Trichotomy law: if r and s are real numbers then one and only one of the statements r < s, r = s and s < r holds.
(2): Transitive law: if r, s, and t are real numbers r < s and s < t and r < t.
(3): If r < s then r+u < s+u for any real number u.
(4): If r < s and u is real number, then ru < su if u > 0.
(5): Completeness property: any nonempty set of numbers that is bounded above has a least upper bound.
The first four properties above are summarized by saying that R is an ordered field. There are other ordered fields. For instance, the rational numbers satisfy (1) to (4) ( reading 'rational' for 'real' each time ), but R is the only ordered field which also has the completeness property (5), i.e. is a complete field. Every nonempty set of real numbers that is bounded below ( has a lower bound ) must have a greatest lower bound.
Question(s):
1: [3/4, 5/6, 7/8, 9/7] is subset of rational number and has 9/7 as least upper bound?
2: Please explain above given definition of 'order properties' in some detail by giving numerical example. Thank you
Sincerely,
vijay
Hi
Please keep your answer simple. Thank you
Note on notation:
<= for "less than or equal" and >= for "greater than or equal" and != as a symbol for "not equal".
Partial Order:
A relation <= between the elements of a set S that satisfies the following three conditions:
1: Reflexive condition: a <= a for each a in S.
2: Antisymmetric condition: for a and b in S, a <= b and b <= a can both hold only if a = b.
3: Transitive condition: if a, b, and c are in S, then a <= b and b <= c together imply a <= c.
If b <= a, then also a >= b; and if a <= b but a != b then a < b. An example of a set with a partial order is the set of natural numbers with n <= m if and only if n divides m.
If every pair of elements a, b in the set is comparable ( i.e. either a <= b or b <= a ) then the partially ordered set ( poset ) is called totally ordered or chain. The set of natural numbers is not totally ordered since, for example, 3 and 5 are not comparable. An example of a totally ordered set is the set of real numbers with the relation <= being the ordinary 'less than or equal to' relation.
I don't understand reflexive condition. I know about reflexive relation though.
Reflexive relation: A relation R on a set A is reflexive if, for all a ( belonging to ) A, a R a. The relation 'identity', for example, is reflexive on the set of natural numbers as every member is identical with itself.
Questions:
1: Can you give me example of any other relation other than 'identity' which is reflexive?
2: What does this ''reflexive condition a <= a for each a in S'' means? Please give me some simple numerical example.
3: What does that mean ''an example of a set with a partial order is the set of natural numbers with n <= m if and only if n divides m.''? What will happen if n does not divide m?
4: What does that mean ''the set of natural numbers is not totally ordered since, for example, 3 and 5 are not comparable''? I think 3 and 5 are comparable and we can write 3<5.
Sincerely,
vijay
Hi
Please keep in mind while answering this post that I'm first year college student so please keep your answers simple. Thank you
Elementary Row and Column operations:
Usually a given system of linear equations is reduced to a simple equivalent system by applying in turn a finite number of elementary operations which are stated below:
1: Interchanging two equations
2: Multiplying an equation by a non-zero number.
3: Adding a multiple of one equation to another equation.
Corresponding to these three elementary operations, the following elementary row operations are applied to matrices to obtain equivalent matrices ( these equivalent matrices still represent the same system of linear equations ). The elementary row operations are:
1: Interchanging two rows
2: Multiplying a row by a non-zero number.
3: Adding a multiple of one row to another row.
There are also elementary column operations which are given below:
1: Interchanging two columns
2: Multiplying a column by a non-zero number.
3: Adding a multiple of one column to another column.
Consider the following system:
x + y + 2z = 1
2x - y +8z = 12
3x + 5y + 4z = -3
which can be written in matrix form as AX=B, where
A = [1,1,2; 2,-1,8; 3,5,4]
X = [x; y; z]
B = [1; 12; -3]
A is called the matrix of coefficients.
Appending a column of constants which is B on the left of A, we get the augmented matrix of given system.
Augmented Matrix = [1,1,2:1; 2,-1,8:12; 3,5,4:-3]
If you see this augmented matrix carefully, it's the the same system of linear equations, each row represents a equation, except that we have omitted the variables and we have colon instead of 'equal to' sign.
Now we can apply the elementary row operations on augmented matrix.
But to apply elementary column operations we have to write this augmented matrix differently, like this:
[1,2,3; 1,-1,5; 2:1,8:12,4:-3]
This augmented matrix still represents the same system of linear equations except that now each column represents a equation.
QUESTIONS:
1: Am I right that while applying elementary column operations we write augmented matrix in such a way that each column represents equation?
2: Don't these row and column operations effect determinant?
Row Echelon Form of Matrix:
We can get a special form of matrix A which is called row echelon form by applying row operations and that row echelon form of matrix A should meet the the following conditions.
1: In each successive non-zero row, the number of zeros before the leading entry ( first non-zero entry in a certain row is known as leading entry ) is greater than the number of zeros in the preceding row.
2: Every non-zero row precedes every zero row ( if any ).
3: The non-zero entry ( leading entry ) in each row is 1.
4 : A is said to be in reduced row echelon form if it is in row echelon form and if the first non-zero entry ( leading entry ) in Row(i) lies in Column(j), then all the other entries of Column(j) are zero.
This fourth condition is used for reduced row echelon form of matrix.
Row echelon form of the this augmented matrix,
Augmented Matrix = [1,1,2:1; 2,-1,8:12; 3,5,4:-3]
is [1,0,0:1; 0,1,0:-2; 0,0,1:1]
QUESTIONS:
1: What are conditions for column row echelon form of matrix?
2: What will be column echelon form this augmented matrix,
Augmented Matrix = [1,2,3; 1,-1,5; 2:1,8:12,4:-3]?
3: Does the concept of echelon form of matrix only applies to augmented matrices?
Row Rank of Matrix:
Let C be a non-zero matrix. If r is the number of non-zero rows when it is reduced to the reduced echelon form, then r is called the row rank of the matrix A.
Reduced echelon form of matrix C,
C = [1,-1,2,-3; 2,0,7,-7; 3,1,12,-11]
is [1,0,7/2,-7/2; 0,1,3/2,-1/2; 0,0,0,0].
So rank of C is 2.
I also tried to find the column rank of C. For that I wrote C differently so that I can apply column operations.
C = [1,2,3; -1,0,1; 2,7,12; -3,-7,-11]
I don't think that C is in column reduced echelon form. But as you can see I can't apply column operation on C any further.
Suspected column echelon form of C is:
[1,0,0; 0,1,0; 7/2,3/2,0; -7/2,-1/2,0]
So I think row rank is also 2.
QUESTIONS:
1: Am I right about the column rank of C?
2: Are row and column ranks of certain matrix always have to be same?
Linearly Independent Rows and Columns:
I think in this matrix [1, 0, 0; 0, 1, 0; 0, 0, 1] all rows are independent. While in matrix [2,-1,1; 1,0,1; 3,-1,2], the third row is not independent because it is the sum of the first two rows.
QUESTIONS:
1: But what's this 'LINEAR' independence?
2: Shall we say that in matrix [2,-1,1; 1,0,1; 3,-1,2], only first two rows are independent?
Please give me some example about column independence.
Sincerely,
Vijay
Hi
We need especially two things in order to proceed with mathematical induction. Steps 1 and 2 determine whether we can proceed or not. Steps 3 and 4 determine accuracy of formula.
1: Formula S(n) -- because without formula we have nothing to prove.
2: Proving formula at n=1 , here 1 is called starting point ( we don't always need to have n=1 as starting point, like in, n! > 2^n, we will start with n=4 because n!>2^n doesn't allow us to use n=1 as starting point ). This step is necessary because if S(n) is not true at n=1 ( or at any other particular starting point n, like n=3,5,6 etc ) then there's really no point in continuing. I think starting with n=1 ( like when formula allows us, formula n! > 2^n doesn't allow us to use n=1 )is convention. Right?
3: Assuming formula is true at n=k.
4: Proving formula at n=k+1. This step decides whether formula is correct or not.
I just came to know that I don't really understand why mathematical induction works. Please keep in mind while answering my questions that I'm going to be first year college student in next few months.
Suppose I just found these three formulae for the sum of n natural numbers (1+2+3+...+n) are:
n(n+1)/2, n(n+1)n!/2, n(n+1)2^(n-1)/2
As I mentioned before we need two things in order to proceed with mathematical induction.
1: Formula -- which we have.
2: Proving formula at n=1 ( if formula allows us - above formulae allow us to use n=1 as starting point )
All three of above formulae are correct at n=1; although I know two of them aren't correct but mathematical induction only needs to evaluate formula at n=1 and making sure it is correct at n=1.
3: In this step we assume above formulae are also correct at some n=k ( k can be any natural number, 50, 60, 70,110 etc ). This step doesn't make any sense to me. We are assuming what we are trying to prove so what's left there to prove.
4: In this step we evaluate above formulae at n=k+1. This step decides the accuracy of formula. But I don't see how.
So how will we know that one of above formulae is correct and other two are inaccurate?
Please don't give me those dominoes kind examples. Thank you
Sincerely,
Vijay
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