Commutativity and Associativity Continued
We know that commutativity holds for the operations $\wedge$ and $\vee$ from the axioms and we also proved associativity holds for these operations. Now that we have some more tools, we can prove some commutativity/associativity properties for the $=$ and $\neq$ operations.
Commutativity
$(a = b) \quad = \quad (b = a)$
\[\begin{align} &a = b && \\ &= (\overline{a} \vee b) \wedge (a \vee \overline{b}) &&\text{Equality} \\ &= (a \vee \overline{b}) \wedge (\overline{a} \vee b) &&\text{Commutativity of }\ \wedge \\ &= (\overline{b} \vee a) \wedge (b \vee \overline{a}) &&\text{Commutativity of }\ \vee \ \text{2 times} \\ &= (b = a) &&\text{Equality} \end{align}\]$(a \neq b) \quad = \quad (b \neq a)$
\[\begin{align} &a \neq b && \\ &= (\overline{a = b}) &&\text{Exclusion} \\ &= (\overline{b = a}) &&\text{Commutativity of } \ = \\ &= (b \neq a) &&\text{Exclusion} \end{align}\]Associativity
$(a = b) = c \quad = \quad a = (b = c)$
\[\begin{align} &(a = b) = c && \\ &= [(\overline{a = b}) \vee c] \wedge [(a = b) \vee \overline{c}] &&\text{Equality} \\ &= [(a \neq b) \vee c] \wedge [(a = b) \vee \overline{c}] &&\text{Exclusion} \\ &= [\color{red}( (a \vee b) \wedge (\overline{a} \vee \overline{b}) \color{red}) \vee c] \wedge [\color{red}( (\overline{a} \vee b) \wedge (a \vee \overline{b}) \color{red}) \vee \overline{c}] &&\text{Equality and Difference} \\ &= [\color{red}( (a \vee b) \vee c \color{red}) \wedge \color{red}( (\overline{a} \vee \overline{b}) \vee c \color{red})] \wedge [\color{red}( (\overline{a} \vee b) \vee \overline{c} \color{red}) \wedge \color{red}( (a \vee \overline{b}) \vee \overline{c} \color{red})] &&\text{Distributivity} \\ &= (a \vee b \vee c) \wedge (\overline{a} \vee \overline{b} \vee c) \wedge (\overline{a} \vee b \vee \overline{c}) \wedge (a \vee \overline{b} \vee \overline{c}) &&\text{Associativity of}\ \wedge \ \text{and} \ \vee\\ &= (\overline{a} \vee \overline{b} \vee c) \wedge (\overline{a} \vee b \vee \overline{c}) \wedge (a \vee b \vee c) \wedge (a \vee \overline{b} \vee \overline{c}) &&\text{Commutativity of}\ \wedge \\ &= [\color{red}( \overline{a} \vee (\overline{b} \vee c) \color{red}) \wedge \color{red}( \overline{a} \vee (b \vee \overline{c}) \color{red})] \wedge [\color{red}( a \vee (b \vee c) \color{red} ) \wedge \color{red}( a \vee (\overline{b} \vee \overline{c}) \color{red})] &&\text{Associativity of}\ \wedge \ \text{and} \ \vee\\ &= [\overline{a} \vee \color{red}( (\overline{b} \vee c) \wedge (b \vee \overline{c}) \color{red})] \wedge [a \vee \color{red}( (b \vee c) \wedge (\overline{b} \vee \overline{c}) \color{red})] &&\text{Distributivity} \\ &= [\overline{a} \vee (b = c)] \wedge [a \vee (b \neq c)] &&\text{Equality and Difference} \\ &= [\overline{a} \vee (b = c)] \wedge [a \vee (\overline{b = c})] &&\text{Exclusion} \\ &= a = (b = c) &&\text{Equality} \end{align}\]$(a = b) \neq c \quad = \quad a = (b \neq c)$
\[\begin{align} &(a = b) \neq c && \\ &= (a = b) = \overline{c} &&\text{Exclusion} \\ &= a = (b = \overline{c}) &&\text{Associativity of =} \\ &= a = (b \neq c) &&\text{Exclusion} \end{align}\]$(a \neq b) = c \quad = \quad a \neq (b = c)$
\[\begin{align} &(a \neq b) = c && \\ &= (\overline{a} = b) = c &&\text{Exclusion} \\ &= \overline{a} = (b = c) &&\text{Associativity of =} \\ &= a \neq (b = c) &&\text{Exclusion} \end{align}\]$(a \neq b) \neq c \quad = \quad a \neq (b \neq c)$
\[\begin{align} &(a \neq b) \neq c && \\ &= (\overline{a = b}) \neq c &&\text{Exclusion} \\ &= \overline{(\overline{a = b})} = c &&\text{Exclusion} \\ &= (a = b) = c &&\text{Double Negation} \\ &= a = (b = c) &&\text{Associativity of =} \\ &= a = \overline{(\overline{b = c})} &&\text{Double Negation} \\ &= a \neq (\overline{b = c}) &&\text{Exclusion} \\ &= a \neq (b \neq c) &&\text{Exclusion} \end{align}\]I find this unintuitive, but you do not need parenthesis around strings of expressions containing the $=$ and the $\neq$ operators. Also interestingly, $(a \neq b \neq c) = (a = b = c)$. It is important to note that $(a \neq b \neq c)$ does not mean what you would think it means. I think many people would want it to mean $(a \neq b) \wedge (b \neq c) \wedge (c \neq a)$. But this is not equivalent and actually that latter statement evaluates to $\color{red}F$.