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Without using truth table, show that (...

Without using truth table, show that
(1) `ptoq-=(p^^q)vv(~p^^~q)`
(2) `~(pvvq)vv(~p^^q)-=~p`

Text Solution

Verified by Experts

LHS=`p""iffq`
`-=(ptoq)^^(qto""p)`
`-=(~pvvq)^^(~qvvp)" "` (Conditional Law)
`-=[~p^^(~qvvp)]vv[q^^(~qvvp)]" "` (Distributive Law)
`-=[(~p^^~q)vv(~p^^p)]vv[(q^^~q)vv(q^^p)]" "` (Distributive Law)
`-=[(~p^^~q)vF]vv(Fvv(q^^p)]" "` (Complement Law)
`-=(~p^^~q)vv(q^^)" "` (Identity Law)
`-=(~p^^~q)vv(p^^q)" "` (Commutative Law)
`-=(p^^q)vv(~p^^~q)" "` (Commutative Law)
=RHS.
(2) `LHS=~(pvvq)vv(~p^^q)`
`-=(~p^^~q)vv(~p^^q)" "` (Negation of disjunction)
`-=~p^^(~qvvq)" "` (Distributive Law)
`-=~p^^T" "` (Complement Law)
`-=~p" "` (Identity Law)
=RHS.
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