1. 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) , Michael P. Frank
(c) , Michael P. Frank

2. 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) , Michael P. Frank
(c) , Michael P. Frank

3. 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) , Michael P. Frank
(c) , Michael P. Frank

4. 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) , Michael P. Frank
(c) , Michael P. Frank

5. 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) , Michael P. Frank
(c) , Michael P. Frank

6. 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) , Michael P. Frank
(c) , Michael P. Frank

7. 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 ( )
11/20/2018
(c) , Michael P. Frank
(c) , Michael P. Frank

8. 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) , Michael P. Frank
(c) , Michael P. Frank

9. 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) , Michael P. Frank
(c) , Michael P. Frank

10. 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) , Michael P. Frank
(c) , Michael P. Frank

11. 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) , Michael P. Frank
(c) , Michael P. Frank

12. Review: Propositional Logic (§§1.1-1.2)
Discrete Math - Module #1 - Logic
11/20/2018
Topic #1 – Propositional Logic
Review: Propositional Logic (§§ )
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) , Michael P. Frank
(c) , Michael P. Frank