List of Logical Equivalences and Truth Tables

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In this article, you will know the list of known logical equivalences and their corresponding truth table as a proof of them being a tautology. To understand more in-depth analysis of each of the identities, you can watch my YouTube channel.

Note that these logical identities are also found in Boolean algebra and each of the logical identity has its dual obtained by inverting the connectives and True or False if any.

Logical Equivalences with And, Or, Not

These are logical equivalence that use following connectives – conjunction, disjunction and negation.

Identity Laws

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\begin{aligned} p \wedge T \equiv p\end{aligned}
pTp \wedge T
TTT
FTF
\begin{aligned}p \vee F \equiv p\end{aligned}
pFp \vee F
TFT
FFF

Domination Laws

\begin{aligned}p \vee T \equiv T\end{aligned}
pTp \vee T
TTT
FTT
\begin{aligned}p \wedge F \equiv F\end{aligned}
pFp \wedge F
TFF
FFF

Idempotent Laws

\begin{aligned}p \wedge p \equiv p\end{aligned}
pp \wedge p
TT
FF
\begin{aligned} p \vee p \equiv p\end{aligned}
pp \vee p
TT
FF

Double Negation Law

\begin{aligned}\neg (\neg p) \equiv p\end{aligned}
p\neg p\neg(\neg p)
TFT
FTF

Commutative Laws

\begin{aligned}p \wedge q \equiv q \wedge p\end{aligned}
pqp \wedge qq \wedge p
TTTT
TFFF
FTFF
FFFF
\begin{aligned}p \vee q \equiv q \vee p\end{aligned}
pqp \vee qq \vee p
TTTT
TFTT
FTTT
FFFF

Associative Laws

\begin{aligned}p \wedge (q \wedge r) \equiv (p \wedge q) \wedge r\end{aligned}
pqrq \wedge rp \wedge (q \wedge r)p \wedge q(p \wedge q) \wedge r
TTTTTTT
TTFFFTF
TFTFFFF
TFFFFFF
FTTTFFF
FTFFFFF
FFTFFFF
FFFFFFF
\begin{aligned}p \vee (q \vee r) \equiv (p \vee q) \vee r\end{aligned}
pqrq \vee rp \vee (q \vee r)p \vee q(p \vee q) \vee r
TTTTTTT
TTFTTTT
TFTTTTT
TFFFTTT
FTTTTTT
FTFTTTT
FFTTTFT
FFFFFFF

Distributive Laws

\begin{aligned} p \wedge (q \vee r) \equiv (p \wedge q) \vee (p \wedge r)\end{aligned}
pqrq \vee rp \wedge (q \vee r)p \wedge qp \wedge r(p \wedge q) \vee (p \wedge r)
TTTTTTTT
TTFTTTFT
TFTTTFTT
TFFFFFFF
FTTTFFFF
FTFTFFFF
FFTFFFFF
FFFFFFFF
\begin{aligned} p \vee (q \wedge r) \equiv (p \vee q) \wedge (p \vee r)\end{aligned}
pqrq \wedge rp \vee (q \wedge r)p \vee qp \vee r(p \vee q) \wedge (p \vee r)
TTTTTTTT
TTFFTTTT
TFTFTTTT
TFFFTTTT
FTTTTTTT
FTFFFTFF
FFTFFFTF
FFFFFFFF

Absorption Laws

\begin{aligned} p \wedge (p \vee q) \equiv p\end{aligned}
pqp \vee qp \wedge (p \vee q)
TTTT
TFTT
FTTF
FFFF
\begin{aligned} p \vee (p \wedge q) \equiv p\end{aligned}
pqp \wedge qp \vee (p \wedge q)
TTTT
TFFT
FTFF
FFFF

Negation Laws

\begin{aligned}p \wedge \neg p \equiv F\end{aligned}
p\neg pp \wedge \neg pF
TFFF
FTFF
\begin{aligned}p \vee \neg p \equiv T\end{aligned}
p\neg pp \vee \neg pT
TFTT
FTTT

De Morgan’s Laws

\begin{aligned}\neg(p \wedge q) \equiv \neg p \vee \neg q\end{aligned}
pq\neg p\neg q(p \wedge q)\neg (p \wedge q)\neg p \vee \neg q
TTFFTFF
TFFTFTT
FTTFFTT
FFTTFTT
\begin{aligned}\neg(p \vee q) \equiv \neg p \wedge \neg q\end{aligned}
pq\neg p\neg q(p \vee q)\neg (p \vee q)\neg p \wedge \neg q
TTFFTFF
TFFTTFF
FTTFTFF
FFTTFTT
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