1. CSE 20: Discrete Mathematics for Computer Science Prof. 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 variable, formula
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. Equivalence Rules

6. Inference Rules:

7. 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

8. Legal steps in proof sequences
Consider the following simplification step (modus ponens)
p(pq)q
Why is it correct to use it?
Because p is necessary for q
Because p is sufficient for q
Because (p(pq))q is a contradiction
Because (p(pq))q is a tautology
None/other

9. Legal steps in proof sequences
A simplification step
(formula 1)(formula 2)
is a correct step, if the formula
(formula 1)(formula 2)
is a tautology
Equivalently, if we assume formula 1 is correct, then we can deduce that formula 2 is correct

10. Applications of proof sequences
Formal verification: a proof that a certain program, or a certain circuit, works correctly
We express the statement “the circuit works correctly” as a formal propositional formula, and then prove it
Typical formulas have millions of inputs, so cannot hope to prove by truth tables. The only way is to find clever ways of building proof sequences

11. 2. Equivalence of Logical Operators
Do we really need IMPLIES and XOR?

12. 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
IFF is necessary

13. 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

14. 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

15. 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

16. 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) p q
P NAND q T F p q
P NOR q T F

17. 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 p q
P NAND q T F

18. 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 p q
P NAND q T F

19. 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 p q
P NAND q T F

20. Can we just use AND, OR?
We saw we can use OR,NOT as primitive connectives – they generate all the rest
Similarly, we can just use NAND
Can we just use AND, OR?
Yes No

21. Monotone predicates
A predicate is monotone if replacing an input variable from F to T can never change the value of the predicate from T to F
Example: AND; OR; At least one out of three
But not: NOT; NAND

22. Monotone predicates
Theorem: a predicate is monotone if and only if it can be described using just AND,OR connectives
We will see the proof in later sessions, after we learn a bit about how to prove theorems
Can you prove it yourself now?