Questions tagged [parity]

This tag is for questions relating to "Parity", a mathematical term that describes the property of an integer's inclusion in one of two categories: even or odd.

In mathematics, parity is a term we use to express if a given integer is even or odd.

  • The parity of a number depends only on its remainder after dividing by $2$.
  • An even number has parity $0$ because the remainder after dividing by $2$ is $0$, while an odd number has parity $1$ because the remainder after dividing by $2$ is $1$.
  • Parity is often useful for verifying whether an equality is true or false by using the parity rules of arithmetic to see whether both sides have the same parity.
  • In information theory, a parity bit appended to a binary number provides the simplest form of error detecting code.
  • Parity is an important idea in quantum mechanics because the wavefunctions which represent particles can behave in different ways upon transformation of the coordinate system which describes them.

References:

https://en.wikipedia.org/wiki/Parity_(mathematics)

http://mathworld.wolfram.com/Parity.html

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Prove that every natural number is either even or odd using induction

It is a pretty basic question at first sight. It seems to be intuitively correct but I can't figure out a way to prove it. I think we have to use strong induction but even after sitting on this question for a long time, I still haven't got any…
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Proving by contradiction

Consider the statement: For all $x, y, z ∈ Z$. At least one of $x−y$, $x−z$ and $y−z$ is even. Prove this statement by contradiction. So the contradictory statement would be that neither one of $x−y$, $x−z$ and $y−z$ is even, eg they're all odd.…
kafo
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Is there a way to test parity of fractional part (only period) of irredecible rational number without calculation?

I search in the web to get any way to test parity of fractional part of irredicible rational number by means to know if that fraction (period) even or odd but i didn't get , for example the fraction part of $\frac {17}{19}$ equal to $…
zeraoulia rafik
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Why should the sum of three numbers be even?

It says in "104 Number Theory Problems" by Titu Andreescu, Dorin Andrica, Zuming Feng (October 25, 2006) in the solution to this problem Example 1.6. Find all positive integers n for which 3n − 4, 4n − 5, and 5n − 3 are all prime…
Micelle
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Parity function definition and intuition, characteristic function of a set.

I have a question of two (and a half) parts relating to parity functions. I) Pertaining to the definition of a parity function. II) Pertaining to the intuition behind checking some specific functions' parity. I) In this particular set of questions,…
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How many digits do I need to determine if the product of a whole number an irrational number is odd or even?

So, say you have a really huge whole number like $5^{2000}$ and an irrational number like $\sqrt(5)$. If you were two multiply the two would you get an even or odd number after rounding to the nearest whole number? $$5^{2000} * \sqrt(5)$$ I realize…
BigBear
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What are linear codes having minimum distance 2 used for?

Consider the following parity check matrix $$H = \begin{bmatrix} 1 & 0 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 0 & 0 & 1 \\ 1 & 1 & 0 & 1 & 0 & 1 \end{bmatrix}$$ Since its 1st and 4th column both are identical, the minimum distance of a code is 2. If we…
Heisenberg
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Is the parity of error function enough to show :$\int_{-l}^{l} \exp ({\operatorname{-x^2erf(x)})dx=\int_{-l}^{l} \exp({\operatorname{x^2erf}}(x)})dx$?

I have tried to show the below identity using the parity of both error function and exp function but I didn't succeed, then my question here is there any analytical way to show this identity or Is the parity of error function enough to show…
zeraoulia rafik
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What is $\{0,1\}^8$?

Recall the parity-bit error detecting method, which uses the function f : B8 → B9 , where B = {0, 1}, defined by f(b1, b2, . . . , b8) = (b1, b2, . . . , b8, b9) where b9 is the sum of the previous 8 bits modulo 2: $b_9 = 0$ if $(b_1+b_2...b_8)$…
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How can I check the parity of transcendental functions?

I know how to check it in general ($f(x)=f(-x)$) but I don't understand how I can check it for any transcendental functions, because I cannot check if (for example) $\tan(x)= \tan(-x)$
Lorenzo B.
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How to convince people that 0 is even

Some people say that 0 is neither even nor odd. I say that 0 is even. Is there a simple way to convince people that 0 is even and the statement that "0 is neither even nor odd" is false.
user172675
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How would I prove that if 11n-5 is odd, then n is an even number using only a direct proof?

I've been able to prove this statement through contraposition and contradiction but I'm struggling to prove it through a direct proof. It seems I always get it in the form where 11n=2(k+3).
DeusDeus
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Prove that $n^2$ is even if and only if $n$ is even

I am practising exam questions and have come across the following. Prove that $n^2$ is even if and only if $n$ is even A contrapositive proof comes to mind, since it must be the case that if $n$ isn't odd then $n$ is even. So $n^2 = (2k+1)^2 =…
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Understanding the received vector in syndrome decoding

I have an exercise, which I do have solutions to but cannot understand the problem text. As far as I understood, syndromes are computed by checking the received vector against the parity check matrix. What is the received vector made out of? $r = c…
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Property of Positive/Negative

Is there a word for whether something has the property of being positive or negative such as "parity" for being odd or even?
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