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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Is odd continuous function differentiable at $x=0$?

Suppose that $f(x)$ is continuous and odd: $f(-x) = - f(x)$. Does it have a derivative at $x=0$? Here is what I got so far: First we calculate $f(0)$ using $f(-0) = -f(0)$, from which $f(0) = 0$. Then we calculate $f'(0)$ as follows: $$ f'(0) =…
Ondřej Čertík
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Seeking proofs that depend on the notion of even-vs-odd parity to prove their points

The notion of the parity is very important in a variety of branches of mathematics. Specifically, I am looking for proofs that use parity in the even-vs-odd sense to prove their points. For example, it is instructive to show that $\sqrt{2}$ is…
Xoque55
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Can decimals/fractions be odd or even?

At school I asked the question and I kept wondering "Can fractions or decimals be odd or even?"
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Knight on a chessboard moving from a1 to h8

I was given a puzzle to solve which goes as:- Can a knight start at square a1 of a chessboard, and go to square h8, visiting each of the remaining squares once on the way ? I reasoned that this won't be possible because both the squares a1 and h8…
kusur
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Symbolic definition for $i$?

Please help! I am trying to find a definition for $i$ that doesn't work for $-i$. let $j$ be either $i$ or $-i$. saying $j^2=-1$ doesn't help since $(i)^2=-1$ and $(-i)^2 = -1$ saying $j=\sqrt{-1}$ doesn't halp since $(-1)^{\frac{1}{2}}$ is…
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Prove the following: If $m$ and $n$ are even integers, then so are $m+n$

I have been asked to prove the following and am having difficulty: If $m$ and $n$ are even integers, then so are $m+n$ and $mn$. My professor has hinted to us to use the definition of an even integer in our proof. This is my proof for $m+n$ thus…
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A closed path is made up of 11 line segments. Can one line, not containing a vertex of the path, intersect each of its segments?

This problem is taken from the book Mathematical Circles by Dmitri Fomin, et al., translated by Mark Saul and published by the American Mathematical Society. Can anyone describe what the question actually means?
Lordinkavu
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Prove that it is not possible to completely cover a 6 × 6 chessboard by tiles which have dimensions 1 × 4.

I think I have some sort of understanding of how to solve this but I'm not sure. I would colour the board with 4 colours such that every 1x4 rectangle would cover one of each colour. Then cover the bottom 4 rows such that we are left with a 2x6…
Mark S
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Erasing numbers from circle and writing down sum

There are $50$ copies of the number $1$, and $50$ copies of the number $-1$, written alternately in a circle. In each step, we pick an arbitrary number, write down the sum of the number and its two neighbors on another piece of paper, and erase that…
Alexi
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Parity and Inverse of Permutations (Odd and Even)

I want an explanation on knowing how to know whether a permutation is odd or even. For example, I have a few permutations of [9] that I need explained for parity, inverse, and number of inversions if possible. 987654321 135792468 259148637 I need…
Salazar
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If $x^2+y^2=z^2$, why can't $x$ and $y$ both be odd?

What does the following mean: If $x^2 + y^2 = z^2$ some integers $z$, then $x$ and $y$ can't be both odd (otherwise, the sum of their squares would be $2$ modulo $4$, which can't be a square). So, one of them must be even? I see that if $x$ and…
user2723
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Parity of the sum involving Stirling numbers of the second kind

How can I decide upon the parity of the expression $N_{n}=\sum_{k=1}^n\sum_{s=0}^k\binom{n}{s}\begin{Bmatrix} n-s\\ k-s \end{Bmatrix}$? Here, $n,k\in \mathbb{N}$ and $\begin{Bmatrix} n\\ k \end{Bmatrix}$ denotes Stirling number of the second kind. I…
MUDASSIR
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Parity of the sum

How can I decide upon the parity of the expression $N_{n}=\sum_{k=1}^n\sum_{s=0}^k\binom{n}{s}\binom{n-s}{k-s}$? Here, $n,k\in \mathbb{N}$. I tried it for $n=1,2,3,4,5,6$. I found that it is odd for $n=2,3,5,6$ and even for $n=1,4$.
MUDASSIR
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Seven vertices of a cube are labeled 0, and the remaining vertex labeled 1. Can you make all labels divisible by 3?

Seven vertices of a cube are labeled 0, and the remaining vertex labeled 1. You’re allowed to change the labels by picking an edge of the cube, and adding 1 to the labels of both of its endpoints. After repeating this multiple times, can you make…
Sunaina Pati
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A visual proof for product of two odd numbers being odd?

Is there a visual proof showing that product of two odd numbers is odd? Or product of a number and an even number is always even? I've got some idea for addition and subtraction. X X X X X + X X X X X X X = [X X] [X X] {X} + [X X] [X X] [X X] {X} =…
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