1 / 14

Enhancing Understanding of Logical Equivalence Proofs

Learn step-by-step equivalence proofs in discrete mathematics using logical operators and known equivalences. Discover how to evolve propositions using logical properties. Explore the need for different logical connectives such as IMPLIES, IFF, and XOR. Experiment with using just one connective - NAND - to simulate others.

clodia
Download Presentation

Enhancing Understanding of Logical Equivalence Proofs

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Peer Instruction in Discrete Mathematics by Cynthia Leeis licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.Based on a work at http://peerinstruction4cs.org.Permissions beyond the scope of this license may be available at http://peerinstruction4cs.org. CSE 20 – Discrete Mathematics Dr. Cynthia Bailey Lee Dr. Shachar Lovett

  2. Today’s Topics: • Step-by-step equivalence proofs • Equivalence of logical operators

  3. 1. Step-by-Step Equivalence Proofs

  4. Two ways to show two propositions are equivalent • Using a truth table • Make a truth table with a column for each • Equivalent iff the T/F values in each row are identical between the two columns • Using known logical equivalences • Step-by-step, proof-style approach • Equivalent iff it is possible to evolve one to the other using only the known logical equivalence properties

  5. Logical Equivalences • (p∧q∧r) ∨ (p∧¬q∧¬r) ∨ (¬p∧q∧r) ∨ (¬p∧¬q∧¬r) ≡(p∧(q∧r)) ∨ (p∧(¬q∧¬r)) ∨ (¬p∧(q∧r)) ∨ (¬p∧(¬q∧¬r)) • ≡ (p∧ ((q∧r) ∨ (¬q∧¬r))) ∨ (¬p ∧ ((q∧r) ∨ (¬q∧¬r))) • Substitute s = (q∧r) ∨ (¬q∧¬r) gives (p∧ s) ∨ (¬p ∧ s) • ≡ (s∧p) ∨ (s∧¬p) • ≡ s ∧ (p ∨ ¬p) • ≡ s ∧ t • ≡ s, substitute back for s gives: • ≡ (q∧r) ∨ (¬q∧¬r) Which law is NOT used? Commutative Associative Distributive Identity DeMorgan’s

  6. 3. Equivalence of Logical Operators Do we really need IMPLIES and XOR?

  7. Are all the logical connectives really necessary? • You already know that IMPLIES is not necessary • p → q≡ ¬p ∨ q • What about IFF? • Replace with ¬(¬p ∨ ¬q) • Replace with (¬p ∧ ¬q) ∨ (p∧q) • Replace with ¬(p∧q) ∧(p∨q) • Replace with something else • XOR is necessary

  8. Are all the logical connectives really necessary? • You already know that IMPLIES is not necessary • p → q≡ ¬p ∨ q • What about XOR? • Replace with ¬(¬p ∨ ¬q) • Replace with (¬p ∧ ¬q) ∨ (p∧q) • Replace with ¬(p∧q) ∧(p∨q) • Replace with something else • XOR is necessary

  9. Are all the logical connectives really necessary? • You already know that IMPLIES is not necessary • p → q≡ ¬p ∨ q • What about AND? • Replace with ¬(¬p ∨ ¬q) • Replace with (¬p ∧ ¬q) ∨ (p∧q) • Replace with ¬(p∧q) ∧(p∨q) • Replace with something else • AND is necessary

  10. Are all the logical connectives really necessary? • Not necessary: • IF • IFF • XOR • AND • We can replicate all these using just two: • OR • NOT • Can we get it down to just one?? YES, just OR YES, just NOT NO, there must be at least 2 connectives Other

  11. It turns out, yes, you can manage with just one! • But it is one we haven’t learned yet: • NAND (NOT AND) • Another option is NOR (NOT OR) • Their truth tables look like this: • Ex: p OR q ≡ (p NAND p) NAND (q NAND q)

  12. Using NAND to simulate the other connectives • “NOT p” is equivalent to • p NAND p • (p NAND p) NAND p • p NAND (p NAND p) • NAND p • None/more/other

  13. Using NAND to simulate the other connectives • “p AND q” is equivalent to • p NAND q • (p NAND p) NAND (q NAND q) • (p NAND q) NAND (p NAND q) • None/more/other

  14. Using NAND to simulate the other connectives • “p OR q” is equivalent to • p NAND q • (p NAND p) NAND (q NAND q) • (p NAND q) NAND (p NAND q) • None/more/other

More Related