Answer to Question #178168 in Discrete Mathematics for Elaine

Question #178168

V. Determine whether each pair of propositions are logically equivalent or not. Use Logical Equivalence Rules.

  1. ¬(p∨(¬p∧q)) and ¬p∧¬q
  2. ¬(p↔q) and p↔¬q
  3. p↔(q∧r) and (p↔q)∧(p↔r)
  4. ¬(p↔q) and p⊕q
1
Expert's answer
2021-04-13T13:42:36-0400

Let us determine whether each pair of propositions are logically equivalent or not. 


1) Since "\u00ac(p\u2228(\u00acp\u2227q))=\u00ac((p\u2228\u00acp)\u2227(p\\lor q)))=\u00ac(T\u2227(p\\lor q))=\u00ac(p\\lor q)=\u00acp\\land \u00acq", we conclude that "\u00ac(p\u2228(\u00acp\u2227q))" and "\u00acp\u2227\u00acq" are logically equivalent.


2) Since "\u00ac(p\u2194q)=\u00ac((p\\to q)\\land (q\\to p))=\u00ac((\\neg p\\lor q)\\land (\\neg q\\lor p))=\n\u00ac(\\neg p\\lor q)\\lor\\neg (\\neg q\\lor p)=(p\\land\\neg q)\\lor (q\\land \\neg p)=(p\\lor q)\\land (\\neg q\\lor\\neg p)=\n(\\neg q\\to p)\\land (p\\to\\neg q)=p\u2194\\neg q", we conclude that "\u00ac(p\u2194q)" and "p\u2194\u00acq" are logically equivalent.


3) Since for "p=r=0, \\ q=1" we have that "p\u2194(q\u2227r)=F\u2194(T\u2227F)=F\u2194F=T" but "(p\u2194q)\u2227(p\u2194r)=(F\u2194T)\u2227(F\u2194F)=F\\land T=F", we conclude that the formulas are not logically equivalent.


4) Since "\u00ac(p\u2194q) = \u00ac((p\\to q)\\land (q\\to p))= \u00ac((\\neg p\\lor q)\\land (\\neg q\\lor p))=\n \u00ac(\\neg( p\\land \\neg q)\\land \\neg( q\\land \\neg p))=\n(p(q\\oplus 1)\\oplus 1)(q(p\\oplus 1)\\oplus 1)\\oplus 1=\n(pq\\oplus p\\oplus 1)(qp\\oplus q\\oplus 1)\\oplus 1=pq\\oplus pq \\oplus pq \\oplus pq \\oplus pq \\oplus p\\oplus pq\\oplus q \\oplus 1\\oplus 1=p\\oplus q,"

we conclude that the formulas "\u00ac(p\u2194q)" and "p\u2295q" are logically equivalent.



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