Laws of Algebra of logical statements - SS2 Mathematics Lesson Note
Let \(p,q,r\) be logical statements, then the following laws hold:
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Commutative Laws:
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\(p \land q = q \land p\)
\(p \vee q = q \vee p\)
Associative Laws:
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\((p \land q) \land r = p \land (q \land r)\)
\((p \vee q) \vee r = p \vee (q \vee r)\)
Distributive Laws:
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\(p \land (q \vee r) = (p \land q) \vee (p \land r)\)
\(p \vee (q \land r) = (p \vee q) \land (p \vee r)\)
Idempotent Laws:
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\(p \land p = p\)
\(p \vee p = p\)
Identity Laws:
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\(p \land T = p\)
\(p \land F = F\)
\(p \vee T = T\)
\(p \vee F = p\)
DeMorgan’s Laws:
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\(\sim(p \land q) = \sim p \vee \sim q\)
\(\sim(p \vee q) = \sim p \land \sim q\)
Complement Laws:
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\(p \land \sim p = F\)
\(p \vee \sim p = T\)
\(\sim(\sim p) = p\)
~T=F; ~F=T
Law of Syllogism (Chain Rule):
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\((p \rightarrow q) \land (q \rightarrow r) \rightarrow (q \rightarrow r)\)