For the following questions you must use the rules of logic (Don’t use truth tables) a) Show that (p harr q) and (not p harr not q ) are logically equivalent. b) Show that not ( p o+ q) and (p harr q) are logically equivalent.

Question
Discrete math
asked 2021-02-19
For the following questions you must use the rules of logic (Don’t use truth tables)
a) Show that (\(\displaystyle{p}\leftrightarrow{q}\)) and (\(\displaystyle\neg{p}\leftrightarrow\neg{q}\) ) are logically equivalent.
b) Show that not ( \(\displaystyle{p}\oplus{q}\)) and (\(\displaystyle{p}\leftrightarrow{q}\)) are logically equivalent.

Answers (1)

2021-02-20
Definitions
\(\displaystyle{p}\leftrightarrow{q}\equiv{\left({p}\wedge{q}\right)}\vee{\left(\neg{p}\wedge\neg{q}\right)}\) (1)
\(\displaystyle{p}\oplus\equiv{\left({p}\vee{q}\right)}\vee\neg{\left({p}\wedge{q}\right)}\) (2)
Double negation law: \(\displaystyle\neg{\left(\neg{p}\right)}\equiv{p}\)
Commutative laws: \(\displaystyle{p}\vee{q}\equiv{q}\vee{p},{p}\wedge{q}\equiv{q}\wedge{p}\)
De Morgan's laws: \(\displaystyle\neg{\left({p}\wedge{q}\right)}\equiv\neg{p}\vee\neg{q},\neg{\left({p}\vee{q}\right)}\equiv\neg{p}\wedge\neg{q}\)
a) \(\displaystyle{\left(\neg{p}\leftrightarrow\neg{q}\right)}\)
\(\displaystyle\equiv{\left(\neg{p}\wedge\neg{q}\right)}\vee{\left(\neg{\left(\neg{p}\right)}\wedge{\left(\neg{\left(\neg{q}\right)}\right)}\right.}\)
\(\displaystyle\equiv{\left(\neg{p}\wedge\neg{q}\right)}\vee{\left({p}\vee{q}\right)}\)
\(\displaystyle\equiv{\left({p}\wedge{q}\right)}\vee{\left(\neg{p}\wedge\neg{q}\right)}\)
\(\displaystyle\equiv{p}\leftrightarrow{q}\)
b) \(\displaystyle\neg{\left({p}\oplus{q}\right)}\)
\(\displaystyle\equiv\neg{\left({\left({p}\vee{q}\right)}\wedge\neg{\left({p}\wedge{q}\right)}\right.}\)
\(\displaystyle\equiv{\left(\neg{\left({p}\vee{q}\right)}\right)}\vee{\left(\neg{\left(\neg{\left({p}\wedge{q}\right)}\right)}\right)}\)
\(\displaystyle\equiv{\left(\neg{\left({p}\vee{q}\right)}\right)}{v}{\left({p}\wedge{q}\right)}\)
\(\displaystyle\equiv{\left({\left(\neg{p}\right)}\wedge{\left(\neg{q}\right)}\right)}\vee{\left({p}\wedge{q}\right)}\)
\(\displaystyle\equiv{\left({p}\wedge{q}\right)}\vee{\left({\left(\neg{p}\wedge{\left(\neg{q}\right)}\right)}\right.}\)
\(\displaystyle\equiv{p}\leftrightarrow{q}\)
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P.vaiue Pevgiue
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