Are the well known operations of logic sum and logic product (defined by their two truth tables) the unique couple of operations (defined on a two elements set) that realize the axioms of: associativity, commutativity, absorption, identity, distributivity, complements?
(The axioms are reported here in section "definition")
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The usual truth value operations (disjunction, conjunction, negation, false, true) constitute the only structure $(\vee, \wedge, \neg, 0, 1)$ that realizes the properties of absorption, complements, commutativity and identity on a two-element set $\{0,1\}$. We show this by proving that the axioms determine the truth tables of $\neg, \vee, \wedge$.
First, observe that by complements $x \vee \neg x = 1$, and by absorption $x \vee 0 = 0$. Thus, if we had $\neg 0 = 0$, we could set $x$ to $0$ and obtain $0 = 0 \vee 0 = 0 \vee \neg 0 = 1$, a contradiction. Similarly, we get $\neg 1 \neq 1$. Thus $\neg$ has to coincide with the usual negation.
By the identity $x \wedge 1 = x$ and commutativity we have $0 \wedge 1 = 1 \wedge 0 = 0$, and $1 \wedge 1 = 1$.
By the identity $x \vee 0 = x$ and commutativity, we have $0 \vee 0 = 0$ and $0 \vee 1 = 1 \vee 0 = 1$.
From here on, we get $0 \wedge 0 = 0 \wedge (0 \vee 0) = 0$ by absorption $x \wedge (x \vee y) = x$, and similarly $1 \vee 1 = 1 \vee (1 \wedge 1) = 1$ by absorption $x \vee (x \wedge y) = x$.
Having determined all values of the functions $\neg, \vee, \wedge$ on $\{0,1\}$, we see that they coincide with the usual truth value operations, as we claimed.