Why is $\wedge$ a minimum and $\vee$ a maximum? 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?
 A: How to remember them?
Long ago someone showed me his method.  I still use it sometimes.
Read the three corners like this:




A: 
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 $p ∧ q$.

