# Can you do boolean and of 1 and a number less than 1? [closed]

where I have encountered these three equations

$$\begin{gather*} e_l=\lvert L-L_l\rvert>T \\ c_\text{max} = \max(c_t, c_r,c_l,c_r,c_{2l}) \\ e_{l'}=e_l \land c_l > 0.5 c_\text{max}. \end{gather*}$$

Because $$e_l$$ is a number being either 0 or 1, and $$c_l$$ is always going to be a fraction number, I am wondering what is the AND result of these two? Would it be either 1 or 0 or it is going to be the value of $$c_l$$ when $$e_l$$ is 1?

• This is the minimum, not the Boolean ‘and’. Sep 25, 2022 at 22:51
• @LSpice Thanks for your reply! Would you mind pointing to a reference telling the operator \land is minimum instead of Boolean "and"? I can't seem to find it anywhere Sep 25, 2022 at 22:54
• The paper you're reading does it … what more would you want for an acceptable reference? It is, for example, quite common in lattice theory. (Note, by the way, that the Boolean ‘and’ becomes a special case of the minimum if we impose the partial order according to which ${\perp} = 0$ is less than $\top = 1$.) Sep 25, 2022 at 22:55
• maybe a page says the meaning of ”^" can mean minimum or something... Sep 25, 2022 at 23:04
• I suspect it could mean "$e_{l'} = e_l$ and $c_l \gt 0.5c_{max}$", i.e. the conjunction of two conditions, not a boolean operation on numbers. But if the context tells you otherwise, then I would guess LSpice's comment is on-target: $\wedge$ also stands for "meet". Sep 25, 2022 at 23:54

The paragraph before says

We calculate the maximum contrast $$c_{max}$$ for all these edges and compare it with the contrast for the left edge [this is $$c_l$$]. If the latter is above a threshold of $$0.5\cdot c_{max}$$ the edge is preserved; otherwise, it is ignored.

So this reads to me like a condition $$c_l \gt 0.5\cdot c_{max}$$.

We also have the definition of $$e_l$$:

a straightforward algorithm would calculate $$e_l=|L−L_l|\gt T$$, where $$e_l$$ is the boolean value that codes whether the edge is active, [emphasis added]

so we should think of $$e_l$$ not as $$0$$ or $$1$$, but a boolean, $$T$$ or $$F$$, giving the truth value of the condition "$$|L−L_l|\gt T$$", which may or may not be true.

Moreover, after he displayed formulas, we have

where $$c_t$$, $$c_r$$, $$c_b$$, $$c_l$$, $$c_{2l}$$ are the contrast deltas for the edges shown in Figure 5, and $$e'_l$$ represents the final boolean value (active or not) for the left edge boundary.

so again, we have that $$e'_l$$ is a boolean, and so the condition $$e_l = e'_l$$ is checking if two booleans are equal (i.e. both true or both false).

The paper is using a symbol, $$\wedge$$, where a word would do, in a false economy, or else insisting on avoiding a word in an equation environment.

• But perhaps we shouldn't call the Boolean $T$ or $F$ when testing $\lvert L - L_l\rvert > T$. 😄 (I've always been partial to the marvellously symmetric $\top$ and $\perp$, although they have reasonable arguments against them. Agreed on the false economy. I was swept away by this marvellous way of expressing ideas as an undergraduate, and I suspect I'm not the only one … but I got over it, as I think most sufferers do!) Sep 27, 2022 at 0:43
• @LSpice oh, I originally wrote $\top$ and $\bot$, but decided to be less fancy and obscure. Sep 27, 2022 at 3:21