Cse 20 discrete mathematics
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CSE 20 – Discrete Mathematics

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Cse 20 discrete mathematics

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


Today s topics

Today’s Topics:

  • Step-by-step equivalence proofs

  • Equivalence of logical operators


1 step by step equivalence proofs

1. Step-by-Step Equivalence Proofs


Two ways to show two propositions are equivalent

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


Logical equivalences

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


3 equivalence of logical operators

3. Equivalence of Logical Operators

Do we really need IMPLIES and XOR?


Are all the logical connectives really necessary

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


Are all the logical connectives really necessary1

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


Are all the logical connectives really necessary2

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


Are all the logical connectives really necessary3

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


It turns out yes you can manage with just one

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)


Using nand to simulate the other connectives

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


Using nand to simulate the other connectives1

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


Using nand to simulate the other connectives2

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


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