Algebraic Properties of Statements

Algebraic Properties of Statements

Algebraic operations are performed among the statements also, but here the operations are the connectives. Let \(p, q\) and \(r\) be three statements. Let \(T\) denotes some true and \(F\) denotes some false statements then the following equivalent statements are valid. Also, with the help of truth tables all of them are proved as follow-

Idempotent Law

• \( p\vee p \equiv p\)

\(\ p \)	\(\ p\vee p\)
\(\ T \)	\(\ T\)
\(\ F \)	\(\ F\)




• \( p\wedge p \equiv p\)

\(\ p \)	\(\ p\wedge p\)
\(\ T \)	\(\ T\)
\(\ F \)	\(\ F\)

Associative Law

• \((p\vee q)\vee r\equiv p\vee (q\vee r) \)

\(\ p \)	\(\ q \)	\(\ r \)	\(\ p\vee{q} \)	\(\ q\vee{r} \)	\(\ (p\vee{q})\vee{r} \)	\(\ p\vee{({q}\vee{r})}\)
\(\ T \)	\(\ T \)	\(\ T \)	\(\ T \)	\(\ T \)	\(\ T \)	\(\ T \)
\(\ T \)	\(\ T \)	\(\ F \)	\(\ T \)	\(\ T \)	\(\ T \)	\(\ T \)
\(\ T \)	\(\ F \)	\(\ T \)	\(\ T \)	\(\ T \)	\(\ T \)	\(\ T \)
\(\ T \)	\(\ F \)	\(\ F \)	\(\ T \)	\(\ F \)	\(\ T \)	\(\ T \)
\(\ F \)	\(\ T \)	\(\ T \)	\(\ T \)	\(\ T \)	\(\ T \)	\(\ T \)
\(\ F \)	\(\ T \)	\(\ F \)	\(\ T \)	\(\ T \)	\(\ T \)	\(\ T \)
\(\ F \)	\(\ F \)	\(\ T \)	\(\ F \)	\(\ T \)	\(\ T \)	\(\ T \)
\(\ F \)	\(\ F \)	\(\ F \)	\(\ F \)	\(\ F \)	\(\ F \)	\(\ F \)




• \((p\wedge q)\wedge r\equiv p\wedge (q\wedge r)\)

\(\ p \)	\(\ q \)	\(\ r \)	\(\ p\wedge{q} \)	\(\ q\wedge{r} \)	\(\ (p\wedge{q})\wedge{r} \)	\(\ p\wedge{({q}\wedge{r})}\)
\(\ T \)	\(\ T \)	\(\ T \)	\(\ T \)	\(\ T \)	\(\ T \)	\(\ T \)
\(\ T \)	\(\ T \)	\(\ F \)	\(\ T \)	\(\ F \)	\(\ F \)	\(\ F \)
\(\ T \)	\(\ F \)	\(\ T \)	\(\ F \)	\(\ F \)	\(\ F \)	\(\ F \)
\(\ T \)	\(\ F \)	\(\ F \)	\(\ F \)	\(\ F \)	\(\ F \)	\(\ F \)
\(\ F \)	\(\ T \)	\(\ T \)	\(\ F \)	\(\ T \)	\(\ F \)	\(\ F \)
\(\ F \)	\(\ T \)	\(\ F \)	\(\ F \)	\(\ F \)	\(\ F \)	\(\ F \)
\(\ F \)	\(\ F \)	\(\ T \)	\(\ F \)	\(\ F \)	\(\ F \)	\(\ F \)
\(\ F \)	\(\ F \)	\(\ F \)	\(\ F \)	\(\ F \)	\(\ F \)	\(\ F \)

Commutative Law

• \(p\vee q \equiv q\vee p\)

\(\ p \)	\(\ q \)	\(\ p \vee{q} \)	\(\ q \vee{p} \)
\(\ T \)	\(\ T \)	\(\ T \)	\(\ T \)
\(\ T \)	\(\ F \)	\(\ T \)	\(\ T \)
\(\ F \)	\(\ T \)	\(\ T \)	\(\ T\)
\(\ F \)	\(\ F \)	\(\ F \)	\(\ F\)




• \(p\wedge q\equiv q\wedge p\)

\(\ p \)	\(\ q \)	\(\ p \wedge{q} \)	\(\ q \wedge{p} \)
\(\ T \)	\(\ T \)	\(\ T \)	\(\ T \)
\(\ T \)	\(\ F \)	\(\ F \)	\(\ F \)
\(\ F \)	\(\ T \)	\(\ F \)	\(\ F\)
\(\ F \)	\(\ F \)	\(\ F \)	\(\ F\)

Distributive Law

• \(p\vee (q\wedge r)\equiv (p\vee q)\wedge (p\vee r) \)

\(\ p \)	\(\ q \)	\(\ r \)	\(\ q\wedge{r} \)	\(\ p\vee{q} \)	\(\ p\vee{r} \)	\(\ p\vee{(q\wedge{r})} \)	\(\ (p\vee{q})\wedge{(p\vee{r})} \)
\(\ T \)	\(\ T \)	\(\ T \)	\(\ T \)	\(\ T \)	\(\ T \)	\(\ T \)	\(\ T \)
\(\ T \)	\(\ T \)	\(\ F \)	\(\ F \)	\(\ T \)	\(\ T \)	\(\ T \)	\(\ T \)
\(\ T \)	\(\ F \)	\(\ T \)	\(\ F \)	\(\ T \)	\(\ T \)	\(\ T \)	\(\ T \)
\(\ T \)	\(\ F \)	\(\ F \)	\(\ F \)	\(\ T \)	\(\ T \)	\(\ T \)	\(\ T \)
\(\ F \)	\(\ T \)	\(\ T \)	\(\ T \)	\(\ T \)	\(\ T \)	\(\ T \)	\(\ T \)
\(\ F \)	\(\ T \)	\(\ F\)	\(\ F \)	\(\ T \)	\(\ F \)	\(\ F \)	\(\ F \)
\(\ F \)	\(\ F \)	\(\ T \)	\(\ F \)	\(\ F \)	\(\ T \)	\(\ F \)	\(\ F \)
\(\ F \)	\(\ F \)	\(\ F \)	\(\ F \)	\(\ F \)	\(\ F \)	\(\ F \)	\(\ F \)




• \( p\wedge (q\vee r)\equiv (p\wedge q)\vee (p\wedge r)\)

\(\ p \)	\(\ q \)	\(\ r \)	\(\ q\vee{r} \)	\(\ p\wedge{q} \)	\(\ p\wedge{r} \)	\(\ p\wedge{(q\vee{r})} \)	\(\ (p\wedge{q})\vee{(p\wedge{r})} \)
\(\ T \)	\(\ T \)	\(\ T \)	\(\ T \)	\(\ T \)	\(\ T \)	\(\ T \)	\(\ T \)
\(\ T \)	\(\ T \)	\(\ F \)	\(\ T \)	\(\ T \)	\(\ F \)	\(\ T \)	\(\ T \)
\(\ T \)	\(\ F \)	\(\ T \)	\(\ T \)	\(\ F \)	\(\ T \)	\(\ T \)	\(\ T \)
\(\ T \)	\(\ F \)	\(\ F \)	\(\ F \)	\(\ F \)	\(\ F \)	\(\ F \)	\(\ F \)
\(\ F \)	\(\ T \)	\(\ T \)	\(\ T \)	\(\ F \)	\(\ F \)	\(\ F \)	\(\ F \)
\(\ F \)	\(\ T \)	\(\ F\)	\(\ T \)	\(\ F \)	\(\ F \)	\(\ F \)	\(\ F \)
\(\ F \)	\(\ F \)	\(\ T \)	\(\ T \)	\(\ F \)	\(\ F \)	\(\ F \)	\(\ F \)
\(\ F \)	\(\ F \)	\(\ F \)	\(\ F \)	\(\ F \)	\(\ F \)	\(\ F \)	\(\ F \)

Identity Law

• \( p\vee F\equiv p\)

\(\ p \)	\(\ F\)	\(\ p\vee F \)
\(\ T \)	\(\ F\)	\(\ T \)
\(\ F \)	\(\ F\)	\(F \)




• \( p\vee T\equiv T\)

\(\ p \)	\(\ T\)	\( p\vee T\)
\(\ T \)	\(\ T\)	\(\ T \)
\(\ F \)	\(\ T\)	\(T \)




• \( p\wedge T \equiv p\)

\(\ p \)	\(\ T\)	\(\ p\wedge T \)
\(\ T \)	\(\ T\)	\(\ T \)
\(\ F \)	\(\ T\)	\(F\)




• \( p\wedge F\equiv F\)

\(\ p \)	\(\ F\)	\(\ p\wedge F \)
\(\ T \)	\(\ F\)	\(\ F \)
\(\ F \)	\(\ F\)	\(F\)

Complement Law

• \( p\vee \sim p\equiv T \)

\(\ p \)	\(\ \sim{p} \)	\(\ p \vee{\sim p} \)	\(\ T \)
\(\ T \)	\(\ F \)	\(\ T \)	\(\ T \)
\(\ F \)	\(\ T \)	\(\ T \)	\(\ T \)




• \( \sim (\sim p)\equiv p \)

\(\ p \)	\(\ \sim{p} \)	\(\ \sim(\sim p) \)
\(\ T \)	\(\ F \)	\(\ T \)
\(\ F \)	\(\ T \)	\(\ F \)




• \( p\wedge \sim p\equiv F\)

\(\ p \)	\(\ \sim{p} \)	\(\ p \wedge{\sim p} \)	\(\ F \)
\(\ T \)	\(\ F \)	\(\ F \)	\(\ F \)
\(\ F \)	\(\ T \)	\(\ F \)	\(\ F \)




• \( \sim T \equiv F\) ; \( \sim F \equiv T\)

\(\ T \)	\(\ \sim{T} \)	\(\ F \)	\(\ \sim{F} \)
\(\ T \)	\(\ F \)	\(\ F \)	\(\ T \)
\(\ T \)	\(\ F \)	\(\ F \)	\(\ T \)

De-Morgan's Law

• \( \sim (p\vee q)\equiv \sim p\wedge \sim q \)

\(\ p \)	\(\ q \)	\(\ \sim{p} \)	\(\ \sim{q} \)	\(\ p\vee{q} \)	\(\sim(p\vee{q}) \)	\(\ \sim{p}\wedge \sim{q} \)
\(\ T\)	\(\ T\)	\(\ F \)	\(\ F \)	\(\ T \)	\(\ F \)	\(\ F \)
\(\ T\)	\(\ F\)	\(\ F \)	\(\ T \)	\(\ T \)	\(\ F\)	\(\ F\)
\(\ F\)	\(\ T\)	\(\ T \)	\(\ F \)	\(\ T \)	\(\ F\)	\(\ F\)
\(\ F\)	\(\ F\)	\(\ T \)	\(\ T \)	\(\ F \)	\(\ T \)	\(\ T \)




• \( \sim (p\wedge q)\equiv \sim p\vee \sim q \)

\(\ p \)	\(\ q \)	\(\ \sim{p} \)	\(\ \sim{q} \)	\(\ p\wedge{q} \)	\(\sim(p\wedge{q}) \)	\(\ \sim{p}\vee \sim{q} \)
\(\ T\)	\(\ T\)	\(\ F \)	\(\ F \)	\(\ T \)	\(\ F \)	\(\ F \)
\(\ T\)	\(\ F\)	\(\ F \)	\(\ T \)	\(\ F \)	\(\ T \)	\(\ T \)
\(\ F\)	\(\ T\)	\(\ T \)	\(\ F \)	\(\ F \)	\(\ T \)	\(\ T \)
\(\ F\)	\(\ F\)	\(\ T \)	\(\ T \)	\(\ F \)	\(\ T \)	\(\ T \)

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