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prove-p-q-and-negetion-of-q-negation-of-p-




Question Number 78671 by berket last updated on 19/Jan/20
prove p⇒q and negetion of q⇒negation of p
$${prove}\:{p}\Rightarrow{q}\:{and}\:{negetion}\:{of}\:{q}\Rightarrow{negation}\:{of}\:{p} \\ $$
Answered by Rio Michael last updated on 19/Jan/20
to prove that p⇒q ≡ q ⇒∼p    1  set up a truth table  p         q        ∼p         ∼q        p⇒q       ∼q ⇒∼p  T         T         F            F          T                   T  T         F         F            T          F                  F  F         T         T            F          T                   T  F         F         T             T         T                   T  we see that the last columns are theresame  thus  p ⇒ q ≡ ∼q ⇒∼p  2 Using D′morgans law     p ⇒ q ≡ ∼( p⇐q)                  ≡ ∼q ⇒ ∼p  also  ∼q ⇒∼p ≡ ∼(q ⇐p)                                 ≡ p ⇒ q   hence they are correct
$${to}\:{prove}\:{that}\:{p}\Rightarrow{q}\:\equiv\:{q}\:\Rightarrow\sim{p} \\ $$$$\:\:\mathrm{1}\:\:\boldsymbol{{set}}\:\boldsymbol{{up}}\:\boldsymbol{{a}}\:\boldsymbol{{truth}}\:\boldsymbol{{table}} \\ $$$${p}\:\:\:\:\:\:\:\:\:{q}\:\:\:\:\:\:\:\:\sim{p}\:\:\:\:\:\:\:\:\:\sim{q}\:\:\:\:\:\:\:\:{p}\Rightarrow{q}\:\:\:\:\:\:\:\sim{q}\:\Rightarrow\sim{p} \\ $$$${T}\:\:\:\:\:\:\:\:\:{T}\:\:\:\:\:\:\:\:\:{F}\:\:\:\:\:\:\:\:\:\:\:\:{F}\:\:\:\:\:\:\:\:\:\:{T}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{T} \\ $$$${T}\:\:\:\:\:\:\:\:\:{F}\:\:\:\:\:\:\:\:\:{F}\:\:\:\:\:\:\:\:\:\:\:\:{T}\:\:\:\:\:\:\:\:\:\:{F}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{F} \\ $$$${F}\:\:\:\:\:\:\:\:\:{T}\:\:\:\:\:\:\:\:\:{T}\:\:\:\:\:\:\:\:\:\:\:\:{F}\:\:\:\:\:\:\:\:\:\:{T}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{T} \\ $$$${F}\:\:\:\:\:\:\:\:\:{F}\:\:\:\:\:\:\:\:\:{T}\:\:\:\:\:\:\:\:\:\:\:\:\:{T}\:\:\:\:\:\:\:\:\:{T}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{T} \\ $$$${we}\:{see}\:{that}\:{the}\:{last}\:{columns}\:{are}\:{theresame} \\ $$$${thus}\:\:{p}\:\Rightarrow\:{q}\:\equiv\:\sim{q}\:\Rightarrow\sim{p} \\ $$$$\mathrm{2}\:\boldsymbol{{Using}}\:\boldsymbol{{D}}'\boldsymbol{{morgans}}\:\boldsymbol{{law}} \\ $$$$\:\:\:{p}\:\Rightarrow\:{q}\:\equiv\:\sim\left(\:{p}\Leftarrow{q}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\equiv\:\sim{q}\:\Rightarrow\:\sim{p} \\ $$$${also}\:\:\sim{q}\:\Rightarrow\sim{p}\:\equiv\:\sim\left({q}\:\Leftarrow{p}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\equiv\:{p}\:\Rightarrow\:{q}\: \\ $$$${hence}\:{they}\:{are}\:{correct} \\ $$$$\:\:\: \\ $$

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