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# x XOR y (decimal integers)

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Look at this in Desmos: https://www.desmos.com/calculator/aenqti0xbv

153 xor 73 = 208

num1 = 15310 = 100110012

num2 = 7310 = 10010012

 num1 1 0 0 1 1 0 0 1 num2 0 1 0 0 1 0 0 1 xor 1 1 0 1 0 0 0 0

11010002 = 20810

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1.  We need to know how the xor bit operator works.
2.  We need to know the binary lengths of x and y.
3.  We need a function that gets an nth digit from the binary form of x and y.

1)

xor table

 x y xor 1 1 0 1 0 1 0 1 1 0 0 0

In math, the xor (bitwise) operator can be written as $$|x-y|$$

2) If we want to know the length of binary x, we can use the function below

$$l(x)=\Bigl\lfloor\dfrac{\operatorname{log}(2x)}{\operatorname{log}(2)}\Bigr\rfloor=\lfloor\operatorname{log}_2x\rfloor+1 = \lceil log_2x\rceil$$

3) To get the nth digit from the binary form of a number, we must divide the number by 2 to the power of n and get the remainder of division by 2.

$$d(x,n)=\Bigl\lfloor\dfrac{x}{2^n}\Bigr\rfloor\operatorname{mod}2$$

$$\Bigl\lfloor\dfrac{x}{2^n}\Bigr\rfloor = \lfloor2^{-n}x\rfloor$$, so

$$d(x,n)=\lfloor2^{-n}x\rfloor\operatorname{mod}2$$

When we "XORing" something we take nth binary digit of x and y and xor them.

$$|d(x,n)-d(y,n)|$$

We do this for all digits of x and y, so it means that we need a summation $$\sum$$
$$f(x,y)=\sum^{\operatorname{max}(l(x),l(y))}_{n=0}|d(x,n)-d(y,n)|$$

max(l(x), l(y)) - means that we begin to count from the highest order of one of two numbers.

Function output is the sum of ones and zeros obtained using the bitwise operator xor.

1. If you want to get decimal output you should to time xor (bitwise) with 2n

$$\operatorname{xor}(x,y)=\sum^{\operatorname{max}(l(x),l(y))}_{n=0}|d(x,n)-d(y,n)|*2^n$$

$$\operatorname{xor}(x,y)=\sum^{\operatorname{max}(l(x),l(y))}_{n=0}|(\lfloor2^{-n}x\rfloor\operatorname{mod}2)-(\lfloor2^{-n}x\rfloor\operatorname{mod}2)|*2^n$$

2. if you want to get binary output you should to time xor (bitwise) with 10n

$$\operatorname{xor}(x,y)=\sum^{\operatorname{max}(l(x),l(y))}_{n=0}|d(x,n)-d(y,n)|*10^n$$

$$\operatorname{xor}(x,y)=\sum^{\operatorname{max}(l(x),l(y))}_{n=0}|(\lfloor2^{-n}x\rfloor\operatorname{mod}2)-(\lfloor2^{-n}x\rfloor\operatorname{mod}2)|*10^n$$

Thanks a lot to heureka!!!! (See the comments below)

All main operators here: https://www.desmos.com/calculator/41d24kmsvd​

off-topic
Oct 3, 2019
edited by JoshuaGreen  Oct 19, 2019
edited by JoshuaGreen  Oct 22, 2019
edited by JoshuaGreen  Oct 22, 2019
edited by JoshuaGreen  Oct 23, 2019

#1
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Mathematical equivalents

see: https://en.wikipedia.org/wiki/Bitwise_operation#OR  Oct 3, 2019

#1
+3

Mathematical equivalents

see: https://en.wikipedia.org/wiki/Bitwise_operation#OR  heureka Oct 3, 2019