logical equivalence - jeremy9959.net
TRANSCRIPT
Equivalence
Two statements P and Q are logically equivalent if P ≈∆ Q is
always true.For example, P =∆ Q and “Q or ≥ P” are logically equivalent.
To see this, look at the truth table for
(P =∆ Q) ≈∆ (≥ P ‚ Q).
We can write “=” instead of ≈∆ in these situations.
t
P I EE EC I
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Other ways of thinking about logical equivalence
Another way to think of logical equivalence is that two statements
made up of statements P, Q, R, . . . are logically equivalent if their
truth tables are the same.
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Boolean Algebra
One can think of a kind of algebra made up of statements together
with operations And, Or, Not, implies, and so on. From this point
of view, two expressions made up of statements are “equal” or “the
same” if they are logically equivalent.
In fact this is called Boolean Algebra and is an important tool in
computer science.
P Q R statements
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DeMorgan’s Laws
I NOT (P AND Q) is equivalent to ((NOT P) or (NOT Q))
I NOT (P OR Q) is equivalent to ((NOT P) and (NOT Q))
prod C PV Q
PAO CPAQ p Q punQPQ FTT T F F F
F T TT F F TF T F T T F T
F F F T T T TT A I
Pva l Ps Q
Contrapositive
(P =∆ Q) = (≥ Q =∆ ≥ P)
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If P then Q If Not On THEN NOT R
P Q P Q Q P Q PT T T F F T
T F F T F F
F T T F T T
F F T T T T
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If it is not cloudy then it is not raining
Associative Laws
P · (Q · R) = (P · Q) · Ra
PVCQUR fPVQ vR
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T F FT T F F
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T T TO Fp
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