Proof by contradiction
Suppose the formula is not a tautology.
Then there exist (p0â,q0â,r0â,s0â)â(T,F)4 such that
âŁ(p0ââqoâ)â(p0ââr0â)â§(q0ââs0â)âr0ââ¨s0ââŁ=F .
The definition of implication implies that
âŁ(p0ââqoâ)â(p0ââr0â)â§(q0ââs0â)âŁ=T and âŁr0ââ¨s0ââŁ=F
The definition of conjunction and disjunction imply that
âŁp0ââqoââŁ=âŁp0ââr0ââŁ=âŁ(q0ââs0â)âŁ=T and âŁr0ââŁ=âŁs0ââŁ=F
It follows from âŁp0ââŁâF=âŁq0ââŁâF=T that âŁp0ââŁ=âŁq0ââŁ=F
Consequently âŁp0ââ¨q0ââŁ=Fâ¨F=F and we have a contradiction with âŁp0ââ¨q0ââŁ=T
Therefore, our assumption is not true, and we conclude that the statement is a TAUTOLOGY
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