Conjunction is denoted in many ways. In programming, it is usually represented by the reserved word and (Pascal, Visual Basic and most languages that are not derived from C), or the symbol && (for most languages derived from C, such as Java, Perl, PHP and etc.).
In electrical and computer engineering, it is represented by a single dot ⋅ (which is also used to indicate multiplication for general algebra). Sometimes, the ⋅ symbol is not even used, the conjunction of A and B is denoted as AB.
In mathematics, it is represented by the ∧ symbol. We shall use the ∧ symbol in this module to represent conjunction.
Formally, ∧⊆ 𝔹 × 𝔹 × 𝔹. This means that conjunction is a relation that is a subset of the Cartesian product of three 𝔹s. In fact, we can examine each element in the relation:
Immediately, we can see that as an operator, conjunction is commutative, as true∧false = false∧true = false. We can also prove that conjunction is associative. This means that (A ∧ B) ∧ C = A ∧ (B ∧ C). The proof involves listing all eight possible combinations of value of A, B and C, and show that the two sides of the equality are indeed the same for all eight combinations.
Last, but not least, true is the identity of conjunction. This is because ∀A ∈ 𝔹 : A ∧ true = A.
Note that conjunction is also a function if we see it as a map from 𝔹 × 𝔹 to 𝔹. As such, the function is not injective, but it is surjective.