Propositional Equivalence (§1.2)

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Propositional Equivalence (§1.2) Discrete Math - Module #1 - Logic 11/20/2018 Topic #1.1 – Propositional Logic: Equivalences Propositional Equivalence (§1.2) Two syntactically (i.e., textually) different compound propositions may be the semantically identical (i.e., have the same meaning). We call them equivalent. Learn: Various equivalence rules or laws. How to prove equivalences using symbolic derivations. 11/20/2018 (c)2001-2004, Michael P. Frank (c)2001-2002, Michael P. Frank

Tautologies and Contradictions Discrete Math - Module #1 - Logic 11/20/2018 Topic #1.1 – Propositional Logic: Equivalences Tautologies and Contradictions A tautology is a compound proposition that is true no matter what the truth values of its atomic propositions are! Ex. p  p [What is its truth table?] A contradiction is a compound proposition that is false no matter what! Ex. p  p [Truth table?] Other compound props. are contingencies. 11/20/2018 (c)2001-2004, Michael P. Frank (c)2001-2002, Michael P. Frank

Discrete Math - Module #1 - Logic 11/20/2018 Topic #1.1 – Propositional Logic: Equivalences Logical Equivalence Compound proposition p is logically equivalent to compound proposition q, written pq, IFF the compound proposition pq is a tautology. Compound propositions p and q are logically equivalent to each other IFF p and q contain the same truth values as each other in all rows of their truth tables. 11/20/2018 (c)2001-2004, Michael P. Frank (c)2001-2002, Michael P. Frank

Proving Equivalence via Truth Tables Discrete Math - Module #1 - Logic 11/20/2018 Topic #1.1 – Propositional Logic: Equivalences Proving Equivalence via Truth Tables Ex. Prove that pq  (p  q). F T T T F T T F F T T F T F T T F F F T 11/20/2018 (c)2001-2004, Michael P. Frank (c)2001-2002, Michael P. Frank

Discrete Math - Module #1 - Logic 11/20/2018 Topic #1.1 – Propositional Logic: Equivalences Equivalence Laws These are similar to the arithmetic identities you may have learned in algebra, but for propositional equivalences instead. They provide a pattern or template that can be used to match all or part of a much more complicated proposition and to find an equivalence for it. 11/20/2018 (c)2001-2004, Michael P. Frank (c)2001-2002, Michael P. Frank

Equivalence Laws - Examples Discrete Math - Module #1 - Logic 11/20/2018 Topic #1.1 – Propositional Logic: Equivalences Equivalence Laws - Examples Identity: pT  p pF  p Domination: pT  T pF  F Idempotent: pp  p pp  p Double negation: p  p Commutative: pq  qp pq  qp Associative: (pq)r  p(qr) (pq)r  p(qr) 11/20/2018 (c)2001-2004, Michael P. Frank (c)2001-2002, Michael P. Frank

Discrete Math - Module #1 - Logic 11/20/2018 Topic #1.1 – Propositional Logic: Equivalences More Equivalence Laws Distributive: p(qr)  (pq)(pr) p(qr)  (pq)(pr) De Morgan’s: (pq)  p  q (pq)  p  q Trivial tautology/contradiction: p  p  T p  p  F Augustus De Morgan (1806-1871) 11/20/2018 (c)2001-2004, Michael P. Frank (c)2001-2002, Michael P. Frank

Defining Operators via Equivalences Discrete Math - Module #1 - Logic 11/20/2018 Topic #1.1 – Propositional Logic: Equivalences Defining Operators via Equivalences Using equivalences, we can define operators in terms of other operators. Exclusive or: pq  (pq)(pq) pq  (pq)(qp) Implies: pq  p  q Biconditional: pq  (pq)  (qp) pq  (pq) 11/20/2018 (c)2001-2004, Michael P. Frank (c)2001-2002, Michael P. Frank

Discrete Math - Module #1 - Logic 11/20/2018 Topic #1.1 – Propositional Logic: Equivalences An Example Problem Check using a symbolic derivation whether (p  q)  (p  r)  p  q  r. (p  q)  (p  r) [Expand definition of ]  (p  q)  (p  r) [Expand defn. of ]  (p  q)  ((p  r)  (p  r)) [DeMorgan’s Law]  (p  q)  ((p  r)  (p  r)) cont. 11/20/2018 (c)2001-2004, Michael P. Frank (c)2001-2002, Michael P. Frank

Discrete Math - Module #1 - Logic 11/20/2018 Topic #1.1 – Propositional Logic: Equivalences Example Continued... (p  q)  ((p  r)  (p  r))  [ commutes]  (q  p)  ((p  r)  (p  r)) [ associative]  q  (p  ((p  r)  (p  r))) [distrib.  over ]  q  (((p  (p  r))  (p  (p  r))) [assoc.]  q  (((p  p)  r)  (p  (p  r))) [trivail taut.]  q  ((T  r)  (p  (p  r))) [domination]  q  (T  (p  (p  r))) [identity]  q  (p  (p  r))  cont. 11/20/2018 (c)2001-2004, Michael P. Frank (c)2001-2002, Michael P. Frank

Discrete Math - Module #1 - Logic 11/20/2018 Topic #1.1 – Propositional Logic: Equivalences End of Long Example q  (p  (p  r)) [DeMorgan’s]  q  (p  (p  r)) [Assoc.]  q  ((p  p)  r) [Idempotent]  q  (p  r) [Assoc.]  (q  p)  r [Commut.]  p  q  r Q.E.D. (quod erat demonstrandum) (Which was to be shown.) 11/20/2018 (c)2001-2004, Michael P. Frank (c)2001-2002, Michael P. Frank

Review: Propositional Logic (§§1.1-1.2) Discrete Math - Module #1 - Logic 11/20/2018 Topic #1 – Propositional Logic Review: Propositional Logic (§§1.1-1.2) Atomic propositions: p, q, r, … Boolean operators:       Compound propositions: s : (p  q)  r Equivalences: pq  (p  q) Proving equivalences using: Truth tables. Symbolic derivations. p  q  r … 11/20/2018 (c)2001-2004, Michael P. Frank (c)2001-2002, Michael P. Frank