1. CS203 Discrete Mathematical Structures
Logic (2)

2. Predicate Logic - everybody loves somebody
Proposition, YES or NO?
3 + 2 = 5
X + 2 = 5
X + 2 = 5 for any choice of X in {1, 2, 3}
X + 2 = 5 for some X in {1, 2, 3}
YES NO
YES
YES

3. … Predicates Ahmed eats pizza at least once a week. Define:
EP(x) = “x eats pizza at least once a week.”
Universe of Discourse - x is a student in CS203
A predicate is a function that takes some variable(s) as arguments and returns True or False.
Note that EP(x) is not a proposition, EP(Hassan) is. …

4. Predicates
A predicate is a property or description of subjects in the universe of discourse. The following predicates are all italicized :
Johnny is tall.
The bridge is structurally sound.
17 is a prime number.
Java: predicates are boolean-valued method calls-
someLinkedList.isEmpty()
isPrime(17)

5. Quantifiers There are two quantifiers Existential Quantifier
“” reads “there exists”
Universal Quantifier
“” reads “for all”
Each is placed in front of a propositional function and binds it to obtain a proposition with semantic value.
Mnemonics:
 -- reverse E signifies “there Exists”
 -- upside-down A signifies “for All”

6. Predicates - the universal quantifier
Suppose P(x) is a predicate on some universe of discourse.
Ex. B(x) = “x is carrying a backpack,” x is set of CS203 students.
The universal quantifier of P(x) is the proposition:
“P(x) is true for all x in the universe of discourse.”
We write it x P(x), and say “for all x, P(x)”
x P(x) is TRUE if P(x) is true for every single x.
x P(x) is FALSE if there is an x for which P(x) is false.
x B(x)?

7. Predicates - the existential quantifier
Suppose P(x) is a predicate on some universe of discourse.
Ex. C(x) = “x has a cat,” x is set of CS203 students.
The existential quantifier of P(x) is the proposition:
“P(x) is true for some x in the universe of discourse.”
We write it x P(x), and say “for some x, P(x)”
x P(x) is TRUE if there is an x for which P(x) is true.
x P(x) is FALSE if P(x) is false for every single x.
x C(x)?

8. Predicates - the existential quantifier
Universe of discourse is people in this room.
B(x) = “x is wearing sneakers.”
L(x) = “x is at least 16 years old.”
Y(x)= “x is less than 24 years old.”
Are either of these propositions true?
x B(x)
x (Y(x)  L(x))
A: only a is true
B: only b is true
C: both are true
D: neither is true

9. Predicates - more examples
Universe of discourse is all creatures.
L(x) = “x is a lion.”
F(x) = “x is fierce.”
C(x) = “x drinks coffee.”
All lions are fierce.
Some lions don’t drink coffee.
Some fierce creatures don’t drink coffee.
x (L(x)  F(x))
x (L(x)  C(x))
x (F(x)  C(x))

10. Predicates - more examples
B(x) = “x is a hummingbird.”
L(x) = “x is a large bird.”
H(x) = “x lives on honey.”
R(x) = “x is richly colored.”
All hummingbirds are richly colored.
No large birds live on honey.
Birds that do not live on honey are dully colored.
Universe of discourse is all creatures.
x (B(x)  R(x))
x (L(x)  H(x))
x (H(x)  R(x))

11. Fundamentals of Logic The Use of Quantifiers Ex universe: real numbers

12. Predicates - the universal quantifier
Universe of discourse is people in this Hall.
B(x) = “x is wearing sneakers.”
L(x) = “x is at least 18 years old.”
Y(x)= “x is less than 24 years old.”
Are either of these propositions true?
x (Y(x)  B(x))
x (Y(x)  L(x))
A: only a is true
B: only b is true
C: both are true
D: neither is true

13. Predicates - quantifier negation
x (L(x)  H(x))
Not all large birds live on honey.
x P(x) means “P(x) is true for every x.”
What about x P(x) ?
Not [“P(x) is true for every x.”]
“There is an x for which P(x) is not true.”
x P(x)
So, x P(x) is the same as x P(x).
x (L(x)  H(x))

14. Predicates - quantifier negation
x (L(x)  H(x))
No large birds live on honey.
x P(x) means “P(x) is true for some x.”
What about x P(x) ?
Not [“P(x) is true for some x.”]
“P(x) is not true for all x.”
x P(x)
So, x P(x) is the same as x P(x).
x (L(x)  H(x))

15. Predicates - quantifier negation
So, x P(x) is the same as x P(x).
So, x P(x) is the same as x P(x).
General rule: to negate a quantifier, move negation to the right, changing quantifiers as you go.

16. Fundamentals of Logic Ex. p(x): x is odd. q(x): x2-1 is even. Negate
(If x is odd, then x2-1 is even.)
There exists an integer x such that x is odd and x2-1 is odd.
(a false statement, the original is true)

17. Predicates - quantifier negation
No large birds live on honey.
x (L(x)  H(x))
 x (L(x)  H(x))
Negation rule
 x (L(x)  H(x))
DeMorgan’s
 x (L(x)  H(x))
Subst for 
What’s wrong with this proof?

18. Fundamentals of Logic
multiple variables

19. Fundamentals of Logic BUT Ex. p(x,y): x+y=17.
: For every integer x, there exists an integer y such that x+y=17. (TRUE)
: There exists an integer y so that for all integer x,
x+y=17. (FALSE)
Therefore,

20. Order matters
Set the universe of discourse to be all natural numbers {0, 1, 2, 3, … }.
Let R (x,y ) = “x < y”.
Q1: What does x y R (x,y ) mean?
Q2: What does y x R (x,y ) mean?

21. Order matters R (x,y ) = “x < y” A1: x y R (x,y ):
“All numbers x admit a bigger number y ”
A2: y x R (x,y ):
“Some number y is bigger than all x”
Q: What’s the true value of each expression?

22. More Practice for Predicate Logic
Nermin likes all movies that Rehab likes (and possibly more).
x [Movie(x)  Likes(Rehab, x)  Likes(Nermin, x)]
There is exactly one AU professor who won a Nobel prize
x[AU_Prof(x)  Wins(x, NobelPrize)]  y,z[y  z  AU_Prof(y)  AU_Prof(z)  Wins(y, NobelPrize)  Wins(z, NobelPrize)]

23. Review of Boolean algebra
Not is a horizontal bar above the number
0 = 1
1 = 0
Or is a plus
0+0 = 0
0+1 = 1
1+0 = 1
1+1 = 1
And is multiplication
0*0 = 0
0*1 = 0
1*0 = 0
1*1 = 1

24. NOT Y = ~X Y = !X Y = not X Y = X’ not(Y,X) Logic Gate: A A’ or -A
(also called an inverter) A
A’ or -A
Truth Table:
Single-throw
Double-pole
Switch: A A -A 1
A’ or -A
Y = ~X Y = !X Y = not X Y = X’ not(Y,X)

25. AND A
Logic Gate:
A*B
Truth Table: B A B
A*B 1 A B
Series Circuit:
A*B

26. AND X & Y (Verilog and ABEL) X and Y (VHDL) X Y X * Y XY (textbook)
and(Z,X,Y) (Verilog) V U

27. OR A Logic Gate: A+B Truth Table: B A B A+B 1 A Parallel Circuit: B1 A
Parallel Circuit: B
A+B

28. OR X | Y (Verilog) X # Y (ABEL) X or Y (VHDL) X + Y (textbook) X V Y
X U Y
or(Z,X,Y) (Verilog)

29. NAND Gate NAND X Y Z 0 0 1 0 1 1 X 1 0 1 1 1 0 Z Y Z = ~(X & Y) X Z Y
Z = ~(X & Y)
nand(Z,X,Y)

30. NOR Gate NOR X Y Z 0 0 1 0 1 0 X 1 0 0 Z 1 1 0 Y Z = ~(X | Y) X Z Y
Z = ~(X | Y)
nor(Z,X,Y)

31. Exclusive-OR Gate XOR X Y Z X Z 0 0 0 Y 0 1 1 1 0 1 1 1 0 Z = X ^ Y
0 0 0 Y
0 1 1
Z = X ^ Y
xor(Z,X,Y)
1 0 1
1 1 0

32. XOR
X ^ Y (Verilog)
X $ Y (ABEL) Y
xor(Z,X,Y) (Verilog)

33. Logic Circuits ≡ Boolean Expressions
All logic circuits are equivalent to Boolean expressions and any boolean expression can be rendered as a logic circuit.
AND-OR logic circuits are equivalent to sum-of-products form.
Consider the following circuits: A
y=aB+Bc
abc B C A B C Y
aBc y Ab
y=abc+aBc+Ab

34. question x+y (x+y)y y Find the output of the following circuit
Answer: (x+y)y
Or (xy)y
x+y
(x+y)y y __

35. question
Write the circuits for the following Boolean algebraic expressions
(x+y)x
_______
x+y
(x+y)x
x+y

36. Draw a circuit diagram for  = (xy' + x'y)z.

39. Let’s compare the resulting circuits
Here are two different but equivalent circuits.
In general the one with fewer gates is “better”:
It costs less to build
It requires less power
But we had to do some work to find the second form