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Discrete mathematical formula types
8)((p? q)→┐(p∨q)

& lt= = & gt((p→q)∧(q→p))→┐(p∨q)

& lt= = & gt┐((┐p∨q)∧(┐q∨p))∨┐(p∨q)

& lt= = & gt(┐(┐p∨q)∨┐(┐q∨p))∨(┐p∧┐q)

& lt= = & gt((┐┐p∧┐q)∨(┐┐q∧┐p))∨(┐p∧┐q)

& lt= = & gt((p∧┐q)∨(q∧┐p))∨(┐p∧┐q)

& lt= = & gt(p∧┐q)∨(┐p∧q)∨(┐p∧┐q)

& lt= = & gtm2∨m 1∨m0,

Therefore, the propositional formula is a satisfiable formula without tautology.

9)((p→q)∧(q→r))→(p→r)

& lt= = & gt┐((┐p∨q)∧(┐q∨r))∨(┐p∨r)

& lt= = & gt(┐(┐p∨q)∨┐(┐q∨r))∨(┐p∨r)

& lt= = & gt((┐┐p∧┐q)∨(┐┐q∧┐r))∨(┐p∨r)

& lt= = & gt(p∧┐q)∨(q∧┐r)∨(┐p∨r)

& lt= = & gt(p∧┐q)∨((q∨(┐p∨r))∧(┐r∨(┐p∨r)))

& lt= = & gt(p∧┐q)∨(┐p∨q∨r)

& lt= = & gt(p∨(┐p∨q∨r))∧(┐q∨(┐p∨q∨r))

& lt= = & gt 1∧ 1

& lt= = & gt 1

So the propositional formula is tautology.