Let $\mathbb{B} = \{0,1\}$ denote the Boolean domain, ordered in the usual way. Then $\mathbb{B}$ is a lattice. It has a join operation $\vee$ that coincides with "OR," a meet operation $\wedge$ that that coincides with "AND," and these distribute over each other. There is also a unique function $\neg : \mathbb{B} \rightarrow \mathbb{B}$ subject to the identities $$x \wedge \neg x = 0, \quad x \vee \neg x = 1.$$
This coincides with "NOT." In other words, $\mathbb{B}$ is a Boolean algebra.

We can define two further binary operations (namely $\rightarrow$ and $\setminus$) on $\mathbb{B}$ as follows, coinciding with IF-THEN and A-BUT-NOT-B.

$$x \rightarrow y = \neg x \vee y, \quad x \setminus y = x \wedge \neg y$$

Each is the negation of the other. But there's a HUGE discrepancy in how important and/or ubiquitous these two operations seem to be:

1. Introductory courses in logic and/or mathematics usually talk at length about IF-THEN, but they almost never mention A-BUT-NOT-B.
2. Software proof assistants almost always have support for the IF-THEN operation, but never (as far as I know) have direct support for the A-BUT-NOT-B operation.
3. Almost every theorem in mathematics is phrased using at least one explicit occurrence of IF-THEN, while almost no theorem is ever phrased using an explicit occurrence of A-BUT-NOT-B. This could be a purely linguistic phenomenon, except that:
4. Most mathematicians seem to think of IF-THEN as an actual "stand-alone" operation, yet at the same time, tend to think of A-BUT-NOT-B as a compound operation, built out of AND and NOT.

> 
> **Question.** How can we explain the discrepancy in importance between $\rightarrow$ (IF-THEN) and $\setminus$ (A-BUT-NOT-B)?
> 
> 
> 

I would be satisfied with an explanation of why we tend to think of mathematics asymmetrically *in general,* as opposed to why the asymmetry exists *in this particular case.*