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8 ≡ 6 (mod 2)

Is this True or False? And why?

Is the (mod 2) applied to both sides of the logical equivalence?

Mauro ALLEGRANZA
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Hamish
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    ${}\pmod2$ is not an operator. It makes no sense to say it is "applied to both sides". Moreover here $\equiv$ denotes congruence, not logical equivalence. $a\equiv b\pmod n$ means that $a-b$ is a multiple of $n$. You may verify that $8-6$ is a multiple of $2$. – Angina Seng Sep 13 '18 at 09:19
  • Logical equivalence is not the only kind of equivalence - the notion also belongs, for example, to the world of equivalence relations. Congruence modulo a positive integer is easily shown to be an equivalence relation. – Mark Bennet Sep 13 '18 at 09:29
  • @LordSharktheUnknown: mod is sometimes an operator, for example in "Let $x=y \mod 2$". This is precisely the source of OP's problem. – TonyK Sep 13 '18 at 09:30
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    @TonyK But it is never an operator when written like this. – Tobias Kildetoft Sep 13 '18 at 09:31
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    Statements such as "Let $x=y\mod2$" are abominations. @TonyK – Angina Seng Sep 13 '18 at 10:25
  • @LordSharktheUnknown: That's as may be. I am not making a value judgment here, I'm just pointing out that it's a genuine source of confusion. – TonyK Sep 13 '18 at 12:29

1 Answers1

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In mathematics, "(mod 2)" is usually not a function. It does not "apply" to a side of an equation. It is a description of the entire equation: that the equation should be taken "(mod 2)".

Depending on the vocabulary you are comfortable with, this means one of the following (equivalent) things:

  • The equation holds "up to multiples of 2".
  • There is a $k$ in the integers, so that adding $2k$ to one side of the equation makes it true.
  • If you subtract one side of the equation from the other, the result is divisible by 2.
  • The equation should be read as being about elements of $\mathbb Z/2\mathbb Z$.

For example, the following are all true: $$ 8 = 0 \pmod2\\ 15 = -9 \pmod 2\\ 0 = 18 \pmod2\\ 20 = 20 \pmod2 $$ To emphasize that these are (usually) not exactly equalities of integers, sometimes additionally the equals symbol $=$ is replaced by another, usually $\equiv$.

Mees de Vries
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