Why is $\wedge$ a minimum and $\vee$ a maximum?

Question

Why does $\wedge$ denote a minimum and $\vee$ a maximum? Where did this notation come from? I keep getting them mixed up because to me, $\wedge$ should be a maximum: it's a hill, or a curve reaching its maximum. Similarly, $\vee$ is a gulf, or a curve reaching its minimum, so it should be minimum. The way I am currently memorizing these notations is actually by using this hill/gulf analogy first, and then quickly reminding myself that it is the opposite of that.

Who decided that it should be this way?

I think they could symbol supremum and infimum. Then it make sense because the supremum is the minimum of the set of upper bounds, and the infimum the maximum of the set of lower bound. Anyway I dont saw this notation, only I see them symbolizing the AND and OR symbols of logic. The minimum comes from below and vice versa, it make sense to me as it too. — Masacroso, Jun 25 '16 at 15:46
What kind of minimum or maximum are you talking about? Minimum/maximum of a set? of a function? — Mike Pierce, Jun 25 '16 at 15:49
I believe (although I can't find the citation for this I remember, so this may be wildly off, and I've converted it from an answer to a comment; still, this can be useful as a mnemonic) they come from the symbols "$\cap$" and "$\cup$" for intersection and union; $\cap$ is $\wedge$ and $\cup$ is $\vee$ in partial orders of sets under inclusion.
As to why "$\cap$" and "$\cup$" are written that way: I believe "$\cup$" is intended to look like a "u" (for union), and $\cap$ is dual to $\cup$. — Noah Schweber, Jun 25 '16 at 15:52
@MikePierce Min/max in the context of partial orders - especially lattices. — Noah Schweber, Jun 25 '16 at 15:52
For those confused by context, OP is likely referring to the join and meet of a lattice. As alluded to below in Noah's answer (now above in Noah's comment), one of the introductory examples of a lattice is of sets ordered by set inclusion, where union and intersection play the role of join and meet respectively. — JMoravitz, Jun 25 '16 at 15:52
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This comment is a bit late, but if you are familiar with intersection and union, I think the following explanation will be perfectly clear; think about the symbols as intersection and union. Let's say you have two numbers, 3 and 5. Draw a 3x5 rectangle. Inside it fits a 3x3 square and outside it fits into a 5x5 square. If you think "How can I intersect/ union two numbers?" The 3x3 square is a great way to intersect 3 and 5, so $3=3\wedge 5,$ and the 5x5 square is a great way to union them, so $5=3\vee 5.$ You can also compare to the intersection and union of {1,2,3} and {1,2,3,4,5}. — Maxim Gilula, Feb 24 '20 at 22:17
For < and > they are mouths eating the bigger item. For ∧ and ∨ they are not eating the bigger item, but the bigger item is always above and the smaller below, so ∨ indicates eating the bigger item above (max) and ∧ indicated eating the smaller item below (min). You must also hear a growly "om nom nom" in you head while reading the symbol too. — jdods, Dec 16 '20 at 17:59
I like @jdods memonic, another one is that you actually do think of the wedge as a hill and think of the things on their sides as its valleys and vice versa. So the minimum is the deeper valley — Felix B., Mar 03 '22 at 12:28

score 35 · Answer 1 · answered Jun 25 '16 at 16:17

35

How to remember them?
Long ago someone showed me his method. I still use it sometimes.
Read the three corners like this:

answered Jun 25 '16 at 16:17

GEdgar

111,679

score 23 · Answer 2 · answered Jun 25 '16 at 17:10

Where did this notation come from?

In lattice theory we have join and meet [see: Helena Rasiowa & Roman Sikorski, The Mathematics of Metamathematics (1963), page 34] :

the least upper bound of $a, b \in A$ will be denoted by $a \cup b$ and called the join of elements $a, b$, and the greatest lower bound of $a, b \in A$ will be denoted by $a \cap b$ and called the meet of $a, b$.

The symbols are motivated by the algebra of sets: the symbols $\cap$ and $\cup$ for intersection and union were used by Giuseppe Peano (1858-1932) in 1888 in Calcolo geometrico secondo l'Ausdehnungslehre di H. Grassmann.

In propositional calculus we have $\lor$ for disjunction, introduced by Russell in manuscripts from 1902-1903 and in 1906 in Russell's paper "The Theory of Implication," in American Journal of Mathematics, vol. 28.

And we have $\land$ for conjunction: first used in 1930 by Arend Heyting in “Die formalen Regeln der intuitionistischen Logik,” Sitzungsberichte der preußischen Akademie der Wissenschaften, phys.-math. Klasse, 1930.

The link is with boolean algebra and its use as interpretation for the propositional calculus:

a Boolean algebra is a non-empty set $A$, together with two binary operations $∧$ and $∨$ (on $A$), a unary operation $'$, and two distinguished elements $0$ and $1$, satisfying the following axioms [...]. There are several possible widely adopted names for the operations $∧, ∨$, and $'$. We shall call them meet, join, and complement (or complementation), respectively. The distinguished elements $0$ and $1$ are called zero and one.

We can then define a binary relation $\le$ in every Boolean algebra; we write $p \le q$ in case $p ∧ q = p$, and we have that:

For each $p$ and $q$, the set $\{ p, q \}$ has the supremum $p ∨ q$ and the infimum $p ∧ q$.

+1. See also this answer at the History of Science and Mathematics site. That answer claims that Pierce introduced $\lor$ in $1865$, apparently referring to this article, but that article isn't quite clear on when and where Peirce used that notation. The collected papers of Peirce are available here, but in a slightly mangled form; I couldn't make heads or tails of them regarding the use of that symbol. — joriki, Jun 25 '16 at 17:21
@joriki - correct ! But the "wedge" used by Peirce and his followerd (at leasr in print: Charles Sanders Peirce (editor), Studies in Logic (1883)) is derived from Boole's $v$ and it is not exactly disjunction. — Mauro ALLEGRANZA, Jun 25 '16 at 17:44
You could also add this almost miracle in notation: when one sees an algebra of sets as a poset or as a thin category the join $\vee$, union $\cup$ and coproduct $\sqcup$ (or $\coprod$) coincide, and similarly $\wedge$, $\cap$ and $\sqcap$ (or $\prod$). — Jackozee Hakkiuz, Jun 29 '20 at 18:24

Why is $\wedge$ a minimum and $\vee$ a maximum?

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