What is symbolic representations of odd and even numbers?

Need to know symbolic representations of odd and even numbers. Its closely related to Number theory.What is symbolic representations of odd and even numbers?
If you want symbols for the sets of all odd or even numbers, you can use 2N+1 and 2N, where N is the set of natural numbers.What is symbolic representations of odd and even numbers?
Evens: {x: x = 2n for all n [an element of] N},





Odds: {x: x = 2n-1 for all n [an element of] N}





The above is written on the basis that N, the set of natural numbers consists of {1, 2, 3, ...}.





Some conventions define N as {0, 1, 2, ...}. With this definition, the set of all odd positive integers would be:





{x: x = 2n+1 for all n [an element of] N}
If you are looking for a symbol that represents the set of all odd numbers or even numbers similar to R = Reals, Q = rationals and N = natural numbers, there does not appear to any in common usage.
This task offers the opportunity for students around CSF level 5 or early CSF level 6 to use symbolic representations of even


and odd numbers, together with various structural principles of algebra, to develop informal proofs, by deduction, of


generalisations developed in Task 3. These principles include the expansion of brackets and grouping like terms.


Other conjectures are offered below as opportunities for students to develop informal proofs. While students will probably need


to specialise these generalisations (and give instances of them in their abstract symbolic form) to interpret them, the intention is


that they develop their ability to develop and express principled arguments.


5.1 Proving the generalisations developed


in Task 3, for example:


even + even = even


odd + odd = even


odd + even = even


even + odd = even.


Also, conjectures about the products:


even 脳 even


odd 脳 odd


odd 脳 even, and the differences:


even 鈥?even


odd 鈥?odd


odd 鈥?even


even 鈥?odd.


Also:


Are the following statements always true


(a and b are positive integers or zero):


If a + b is even then so is a 鈥?b?


Is the converse of this statement true?


If a + b is even and a 鈥?b is even then ab


is even?


The product of two consecutive whole


numbers is even?


(Also see the extensions for this Task over


the page).


Use of abstractions, algebraic


principles and instantiations (as


needed) as tools to develop


principled arguments using the


processes of Specialising, Correcting,


Generalising, Explaining, Justifying,


and Convincing.


For example:


Prove the statement that the sum of


an odd and even number always


gives an odd number.


Let the odd number be 2n + 1


(n is zero or a positive integer).


Let the even number be 2m (m


is zero or a positive integer).


So, the sum of the odd and even


number may be represented as:


2n + 1 + 2m.


This can be restated as:


2n + 2m + 1, and again as:


2(n + m) + 1.


Now, 2 (n + m) must be even


(we have shown earlier that


doubling any number yields an


even number).


So, 鈥?(n + m) + 1鈥?represents an


even number plus one. We have


shown earlier that one more than


an even number is always odd.


Thus,


2(n + m) + 1 is odd, and the


statement that 鈥榯he sum of an


odd and even number always


gives an odd number鈥?is always


true.


Reasoning and strategies


MARSR 501, 502 through to 601,


602 and 603


MARSS 501, 502, 503, 504


through to 601, 602 and 603, 604


Number


As for Task 4


Algebra


MAALE 501, 502, 503 through to


601, 602, 603, 604 and 605, 606


Engagement Links to outcomes in the


Mathematics CSF and


Resources


Interpretations about the


ways of mathematical


thinking involved





You should open these two links you will get a lot about your answer


http://www.vcaa.vic.edu.au/prep10/csf/pu鈥?/a>


http://resources.mhs.vic.edu.au/fs1sl/CS鈥?/a>
2n for evens, 2n+1 for odds. not satisfied?


i use them a lot :)