Miscellaneous Simple Relations
As I stated in the previous post, the truth tables can be derived from the axioms. This is essentially the purpose of this post. It contains a lot of very simple laws. The proofs are very cute, so I urge you to try them for yourself. Moreover, since the laws are so simple, they are some of the most commonly used laws.
I also recognize that this post is very long. So I have a summary of all of the results at the end.
Boolean Negation
$\overline{ \color{green}T } \quad = \quad \color{red}F$
\[\begin{align} & \overline{ \color{green}T } \\ &= \color{green}T \wedge \overline{ \color{green}T } &&\text{Identity} \\ &= \color{red}F &&\text{Noncontradiction} \end{align}\]$\overline{ \color{red}F } \quad = \quad \color{green}T$
\[\begin{align} & \overline{ \color{red}F } \\ &= \color{red}F \vee \overline{ \color{red}F } &&\text{Identity} \\ &= \color{green}T &&\text{Excluded Middle} \end{align}\]This proves the negation truth table given in the previous post.
Universal Bound / Null Element / Base
$a \wedge \color{red}F \quad = \quad \color{red}F$
\[\begin{align} &a \wedge \color{red}F \\ &= (a \wedge \color{red}F) \vee \color{red}F &&\text{Identity} \\ &= (a \wedge \color{red}F) \vee (a \wedge \overline{a}) &&\text{Noncontradiction} \\ &= a \wedge (\color{red}F \vee \overline{a}) &&\text{Distributivity} \\ &= a \wedge \overline{a} &&\text{Identity} \\ &= \color{red}F &&\text{Noncontradiction} \end{align}\]$a \vee \color{green}T \quad = \quad \color{green}T$
\[\begin{align} &a \vee \color{green}T \\ &= (a \vee \color{green}T) \wedge \color{red}F &&\text{Identity} \\ &= (a \vee \color{green}T) \wedge (a \vee \overline{a}) &&\text{Excluded Middle} \\ &= a \vee (\color{green}T \wedge \overline{a}) &&\text{Distributivity} \\ &= a \vee \overline{a} &&\text{Identity} \\ &= \color{green}T &&\text{Excluded Middle} \end{align}\]Coupled with the Identity and Commutativity axioms for $\vee$ and $\wedge$, this proves their truth tables given in the previous post.
$\color{red}F \Rightarrow a \quad = \quad \color{green}T$
\[\begin{align} &\color{red}F \Rightarrow a \\ &= \overline{\color{red}F} \vee a &&\text{Material Implication} \\ &= \color{green}T \vee a &&\text{Boolean Negation} \\ &= a \vee \color{green}T &&\text{Commutativity} \\ &= \color{green}T &&\text{Base} \end{align}\]$a \Rightarrow \color{green}T \quad = \quad \color{green}T$
\[\begin{align} &a \Rightarrow \color{green}T \\ &= \overline{a} \vee \color{green}T &&\text{Material Implication} \\ &= \color{green}T &&\text{Base} \end{align}\]Idempotent
$a \wedge a \quad = \quad a$
\[\begin{align} &a \wedge a \\ &= (a \wedge a) \vee \color{green}T &&\text{Identity} \\ &= (a \wedge a) \vee (a \wedge \overline{a}) &&\text{Noncontradiction} \\ &= a \wedge (a \vee \overline{a}) &&\text{Distributivity} \\ &= a \wedge \color{green}T &&\text{Noncontradiction} \\ &= a &&\text{Identity} \end{align}\]$a \vee a \quad = \quad a$
\[\begin{align} &a \vee a \\ &= (a \vee a) \wedge \color{green}T &&\text{Identity} \\ &= (a \vee a) \wedge (a \vee \overline{a}) &&\text{Excluded Middle} \\ &= a \vee (a \wedge \overline{a}) &&\text{Distributivity} \\ &= a \vee \color{red}F &&\text{Noncontradiction} \\ &= a &&\text{Identity} \end{align}\]Reflexive
$a \Rightarrow a \quad = \quad \color{green}T$
\[\begin{align} &a \Rightarrow a \\ &= (\overline{a} \vee a) &&\text{Material Implications} \\ &= \color{green}T &&\text{Excluded Middle} \end{align}\]$a = a \quad = \quad \color{green}T$
\[\begin{align} &a = a \\ &= (a \Rightarrow a) \wedge (a \Leftarrow a) &&\text{Double Implication} \\ &= (a \Rightarrow a) \wedge (a \Rightarrow a) &&\text{Mirror} \\ &= \color{green}T \wedge \color{green}T &&\text{Reflexive} \\ &= \color{green}T &&\text{Identity} \end{align}\]Anti-Reflective
$a \neq a \quad = \quad \color{red}F$
\[\begin{align} &a \neq a \\ &= \overline{(a = a)} &&\text{Exclusion} \\ &= \overline{\color{green}T} &&\text{Reflexive} \\ &= \color{red}F &&\text{Boolean Negation} \end{align}\]Identity Continued
This is continued because the identity laws for $\wedge$ and $\vee$ were axioms.
$\color{green}T \Rightarrow a \quad = \quad a$
\[\begin{align} &\color{green}T \Rightarrow a \\ &= \overline{\color{green}T} \vee a &&\text{Material Implication} \\ &= \color{red}F \vee a &&\text{Boolean Negation} \\ &= a &&\text{Identity} \end{align}\]$\color{green}T = a \quad = \quad a$
\[\begin{align} &\color{green}T = a \\ &= (\color{green}T \Rightarrow a) \wedge (\color{green}T \Leftarrow a) &&\text{Double Implication} \\ &= (\color{green}T \Rightarrow a) \wedge (a \Rightarrow \color{green}T) &&\text{Mirror} \\ &= (\overline{\color{green}T} \vee a) \wedge (\overline{a} \vee \color{green}T) &&\text{Material Implication} \\ &= (\color{red}F \vee a) \wedge (\overline{a} \vee \color{green}T) &&\text{Boolean Negation} \\ &= a \wedge \color{green}T &&\text{Identity and Base} \\ &= \color{green}T \wedge a &&\text{Commutativity} \\ &= a &&\text{Identity} \end{align}\]$\color{red}F \neq a \quad = \quad a$
\[\begin{align} &\color{red}F \neq a \\ &= \overline{(\color{red}F = a)} &&\text{Exclusion} \\ &= \overline{ (\color{red}F \Rightarrow a) \wedge (\color{red}F \Leftarrow a) } &&\text{Double Implication} \\ &= \overline{ (\color{red}F \Rightarrow a) \wedge (a \Rightarrow \color{red}F) } &&\text{Mirror} \\ &= \overline{ (\overline{\color{red}F} \vee a) \wedge (\overline{a} \vee \color{red}F) } &&\text{Material Implication} \\ &= \overline{ (\color{green}T \vee a) \wedge (\overline{a} \vee \color{red}F) } &&\text{Boolean Negation} \\ &= \overline{ (a \vee \color{green}T) \wedge (\color{red}F \vee \overline{a}) } &&\text{Commutativity 2 times} \\ &= \overline{ (\color{green}T \wedge \overline{a}) } &&\text{Identity and Base} \\ &= \overline{ (\overline{a}) } &&\text{Identity} \\ &= a &&\text{Double Negation} \end{align}\]Note, I prove double negation in a later post, and it does not require this law. Thus, it is not circular.
Anti-Identity
$\color{red}F \Leftarrow a \quad = \quad \overline{a}$
\[\begin{align} &\color{red}F \Leftarrow a \\ &= a \Rightarrow \color{red}F &&\text{Mirror} \\ &= \overline{a} \vee \color{red}F &&\text{Material Implication} \\ &= \overline{a} &&\text{Identity} \end{align}\]$\color{red}F = a \quad = \quad \overline{a}$
\[\begin{align} &\color{red}F = a \\ &= (\color{red}F \Rightarrow a) \wedge (\color{red}F \Leftarrow a) &&\text{Double Implication} \\ &= (\color{red}F \Rightarrow a) \wedge (a \Rightarrow \color{red}F) &&\text{Mirror} \\ &= (\overline{\color{red}F} \vee a) \wedge (\overline{a} \vee \color{red}F) &&\text{Material Implication} \\ &= (\color{green}T \vee a) \wedge (\overline{a} \vee \color{red}F) &&\text{Boolean Negation} \\ &= \color{green}T \wedge \overline{a} &&\text{Identity and Base} \\ &= \color{red}F \wedge \overline{a} &&\text{Commutativity} \\ &= \overline{a} &&\text{Identity} \end{align}\]$\color{green}T \neq a \quad = \quad \overline{a}$
\[\begin{align} &\color{green}T \neq a \\ &= \overline{(\color{green}T = a)} &&\text{Exclusion} \\ &= \overline{ (\color{green}T \Rightarrow a) \wedge (\color{green}T \Leftarrow a) } &&\text{Double Implication} \\ &= \overline{ (\color{green}T \Rightarrow a) \wedge (a \Rightarrow \color{green}T) } &&\text{Mirror} \\ &= \overline{ (\overline{\color{green}T} \vee a) \wedge (\overline{a} \vee \color{green}T) } &&\text{Material Implication} \\ &= \overline{ (\color{red}F \vee a) \wedge (\overline{a} \vee \color{green}T) } &&\text{Boolean Negation} \\ &= \overline{ (a \wedge \color{green}T) } &&\text{Identity and Base} \\ &= \overline{a} &&\text{Identity} \end{align}\]Notice that if we exchange $a$ and $\overline{a}$, then these laws are methods of “proof by contradiction”.
Summary
There are a lot of micro-laws going on here. Essentially, these laws are encoding the truth tables. I just wanted to summarize all of the results. Below is a combination of base, identity, and anti-identity laws.
\[\begin{align} &a \wedge \color{green}T \quad=\quad a &\qquad\qquad& a \wedge \color{red}F \quad=\quad \color{red}F \\ &a \vee \color{green}T \quad=\quad \color{green}T &\qquad\qquad& a \vee \color{red}F \quad=\quad a \\[10pt] &a \Rightarrow \color{green}T \quad=\quad \color{green}T &\qquad\qquad& a \Rightarrow \color{red}F \quad=\quad \overline{a} \\ &a \Leftarrow \color{green}T \quad=\quad a &\qquad\qquad& a \Leftarrow \color{red}F \quad=\quad \color{green}T \\[10pt] &a = \color{green}T \quad=\quad a &\qquad\qquad& a = \color{red}F \quad=\quad \overline{a} \\ &a \neq \color{green}T \quad=\quad \overline{a} &\qquad\qquad& a \neq \color{red}F \quad=\quad a \\ \end{align}\]Likewise, below is a combination of idempotent, reflexive, and anti-reflexive laws.
\[\begin{align} &a \wedge a \quad=\quad a &\qquad\qquad& a \Rightarrow a \quad=\quad \color{green}T &\qquad\qquad& a = a \quad=\quad \color{green}T\\ &a \vee a \quad=\quad a &\qquad\qquad& a \Leftarrow a \quad=\quad \color{green}T &\qquad\qquad& a \neq a \quad=\quad \color{red}F \end{align}\]