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Proofs Using Logical Equivalences

Proofs Using Logical Equivalences. Rosen 1.2. List of Logical Equivalences. p T  p; pF  p Identity Laws pT  T; pF  F Domination Laws pp  p; pp  p Idempotent Laws (p)  p Double Negation Law pq  qp; pq  qp Commutative Laws

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Proofs Using Logical Equivalences

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  1. Proofs Using Logical Equivalences Rosen 1.2

  2. List of Logical Equivalences pT  p; pF  p Identity Laws pT  T; pF  F Domination Laws pp  p; pp  p Idempotent Laws (p)  p Double Negation Law pq  qp; pq  qp Commutative Laws (pq) r  p (qr); (pq)  r  p  (qr) Associative Laws

  3. List of Equivalences p(qr)  (pq)(pr) Distribution Laws p(qr)  (pq)(pr) (pq)(p  q) De Morgan’s Laws (pq)(p  q) Miscellaneous p  p  T Or Tautology p p  F And Contradiction (pq)  (p  q) Implication Equivalence pq(pq)  (qp) Biconditional Equivalence

  4. The Proof Process Assumptions -Definitions -Already-proved equivalences -Statements (e.g., arithmetic or algebraic) Logical Steps Conclusion (That which was to be proved)

  5. Prove: (pq)  q  pq (pq)  q Left-Hand Statement  q  (pq) Commutative  (qp)  (q q) Distributive  (qp)  T Or Tautology  qp Identity  pq Commutative Begin with exactly the left-hand side statement End with exactly what is on the right Justify EVERY step with a logical equivalence

  6. Prove: (pq)  q  pq (pq)  q Left-Hand Statement  q  (pq) Commutative • (qp)  (q q) Distributive Why did we need this step? Our logical equivalence specified that  is distributive on the right. This does not guarantee the equivalence works on the left! Ex.: Matrix multiplication is not always commutative (Note that whether or not is distributive on the left is not the point here.)

  7. Prove: p  q  q  p Contrapositive p  q  p  q Implication Equivalence  q  p Commutative  (q)  p Double Negation  q  p Implication Equivalence

  8. Prove: p  p  q is a tautologyMust show that the statement is true for any value of p,q. p  p  q  p  (p  q) Implication Equivalence  (p  p)  q Associative  (p  p)  q Commutative  T  q Or Tautology  q  T Commutative • T Domination This tautology is called the addition rule of inference.

  9. Why do I have to justify everything? • Note that your operation must have the same order of operands as the rule you quote unless you have already proven (and cite the proof) that order is not important. • 3+4 = 4+3 • 3/4  4/3 • A*B  B*A for everything (for example, matrix multiplication)

  10. Prove: (pq)  p is a tautology (pq)  p  (pq)  p Implication Equivalence  (pq)  p DeMorgan’s  (qp)  p Commutative  q (p  p) Associative  q (p  p) Commutative  q T Or Tautology  T Domination

  11. Prove or Disprove p  q  p  q ??? To prove that something is not true it is enough to provide one counter-example. (Something that is true must be true in every case.) p q pq pq F T T F The statements are not logically equivalent

  12. Prove:p  qp  q p  q  (pq)  (qp) Biconditional Equivalence  (pq)  (qp) Implication Equivalence (x2)  (pq)  (qp) Double Negation  (qp)  (pq) Commutative  (qp)  (pq) Double Negation  (qp)  (pq) Implication Equivalence (x2)  p  q Biconditional Equivalence

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