# logic symbol for 'unlike, differing from'

I'd like to express 'unlike, differing from', in an 'academic/professional' fashion using logic symbol(s) or mathematical operators. The descriptions of the corresponding Unicode block seem to show that there's not one symbol for it.

I guess it is related with formal semantics/linguistics.

Example: Unlike set B, set A contains X (or X belongs to A).

• I'm assuming OP is looking for a mathematical symbol and has tried finding one by checking what uniciode symbols there are for math stuff (but hasn't found anything yet). Jan 27 '17 at 21:48
• If you're not working specifically in mathematical logic or formal semantics, it's preferable not to use such symbols. Professional mathematicians generally avoid symbols like the universal quantifier, existential quantifier, conjunction, disjunction, etc. unless they're working on logic. They write out these concepts in words: for all, for some, and, or, etc. Jan 28 '17 at 2:02
• Unlike and differing from are not the same relation; and both of them require some standard(s) to be compared, and a mode of comparison. There is no way to translate "Unlike A, B" into logic because unlike is not a functor. Jan 28 '17 at 3:33
• @curoiusdannii I share your concern about single-word questions, but in this case I think it's a positive contribution to the site in that it's a mostly language-independent (= linguistic rather than philological) question. Jan 29 '17 at 0:21
• Excuse my naïveness, but, apparently, in view of the example you offer, what you are after is not a formal symbol expressing the two-place relation of 'unlikeness' or 'difference' (otherwise what would be wrong with the non-identity symbol "≠" ?) but one that expresses a THREE-place relation of unlikeness/difference between x and y IN RESPECT z, is that right? If so, as far as I know, there isn´t such a unique symbol; you will have to use some 'paraphrasis' in one of the logical languages (depending on the nature of x, y and z, entities, sets, etc.).
– user6814
Jan 29 '17 at 18:55

(Why is there still no MathJax support for this SE?! Googling and copy-pasting unicode symbols every time you want to talk about semantics is really annoying.)

What you want to say is basically

¬φ(a) ∧ φ(b)


("The satement φ involving a is not true, but it is true involving b" = "Unlike a, statement φ applies to b).
Note that my a corresponds to your set B and vice versa.
Note also that φ(a) is not meant to denote a one-place predicate φ applyig to some individual a - in this case, convention would be to use the letters P, Q, ... to denote such predicates, and you would not either use an individual constant a to denote a set - but simply an abbreviation for some term a occcuring somewhere in a formula φ.

Let's tackle the question with propositional logic first:

There is a rarely used symbol ↚ (an arrow from right to left with a strike-through, reflecting the negation of a backwards implication →) expressing that relation. You could use that one; however, as expemlified by the two truth tables below, this very uncommon symbol can be reduced to simply ¬p ∧ q:

p ↚ q  ¬p ∧ q
1 0 1  01 0 1
1 0 0  01 0 0
0 1 1  10 1 1
0 0 0  10 0 0
*       *


Applied to your example, p would denote "the property applies to set B", while q would denote "the porperty applies to set A". As you can see, the statement only gets true if p is false and q is true. (The colums with the star below contain the truth values of the final expression.) If you compare these two columns, it gets obvious that the two statements are equivalent (= have the same permutation of truth values, namely 0 0 1 0).

So you could use p ↚ q to abbreviate "p is not true, but q is".

Going back to predicate logic, using this relation, you can express

Unlike set B, set A contains x

as

B∍x ↚ A∍x


which is equivalent to

B does not contain x, and A does contain x

¬(B∍x) ∧ A∍x


If you want to swap sides, you can use the reverse arrow

A contains x, unlike B

A∍x ↛ B∍x


This should do the job for you.

But again, it's way simpler to just write

A∍x ∧ ¬(B∍x)


meaning

A contains x, and B doesn't.