# Why is $\wedge$ a minimum and $\vee$ a maximum? [closed]

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. Jun 25, 2016 at 15:46
• 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 $\cup$. Jun 25, 2016 at 15:52
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• i think of logic/programming: 0 and 1 is 0, 0 or 1 is 1. Jan 24, 2019 at 9:58
• 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}. Feb 24, 2020 at 22:17

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

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. Jun 25, 2016 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. Jun 25, 2016 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 $\prod$). Jun 29, 2020 at 18:24