Logic: Truth Tables, DeMorgan’s Laws
Contemporary Math
Josh Engwer
TTU
20 July 2015
Josh Engwer (TTU)
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
1 / 59
Truth Tables for the AND, OR, NOT Connectives
P Q
T T
T F
F T
F F
Truth Table for Conjunction (AND):
Truth Table for Disjunction (OR):
Truth Table for Negation (NOT):
P Q
T T
T F
F T
F F
P
T
F
P∧Q
T
F
F
F
P∨Q
T
T
T
F
∼P
F
T
(Truth tables for the Conditional & Biconditional are seen in next section.)
Josh Engwer (TTU)
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
2 / 59
Inclusive OR (∨) versus Exclusive OR (XOR)
Disjunction is implicitly inclusive OR.
P
T
The truth table for inclusive OR: T
F
F
Q
T
F
T
F
P
T
The truth table for exclusive OR, is: T
F
F
P∨Q
T
T
T
F
Q
T
F
T
F
P XOR Q
F
T
T
F
The difference in their truth tables is in blue.
Examples in English:
Inclusive OR: ”The car is compact or red” (or both compact and red)
Exclusive OR: ”I (either) drove to Austin or drove to Dallas” (but not both)
Josh Engwer (TTU)
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
3 / 59
Logic Connectives (Order of Operations)
It’s important to know the ”order of operations” of logic connectives.
Otherwise, statements would require too many parentheses & brackets.
DOMINANCE:
MOST DOMINANT
2nd DOMINANT
CONNECTIVES:
Biconditional ←→
Conditional −→
Conjunction ∧
rd
3 DOMINANT
Disjunction ∨
LEAST DOMINANT
Negation
∼
REMARK: Since conjunction & disjunction has equal dominance, statements
involving several of them require parentheses & square brackets!
For example:
P ∧ Q ∨ R is ambiguous! It needs to change to one of the following:
? (P ∧ Q) ∨ R
? P ∧ (Q ∨ R)
? WARNING: The above two statements have different truth tables!
(∼ P ∨ ∼ Q) ∧ ∼ R is equivalent to [(∼ P) ∨ (∼ Q)] ∧ (∼ R)
(Q ∧ ∼ P) −→ ∼ R is equivalent to [(Q ∧ (∼ P))] −→ (∼ R)
Josh Engwer (TTU)
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
4 / 59
Truth Tables (Example)
WEX 3-2-1: Construct a truth table for the logic statement: ∼ (P ∧ ∼ Q)
Josh Engwer (TTU)
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
5 / 59
Truth Tables (Example)
WEX 3-2-1: Construct a truth table for the logic statement: ∼ (P ∧ ∼ Q)
P Q
Josh Engwer (TTU)
∼Q P∧∼Q
∼ (P ∧ ∼ Q)
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
6 / 59
Truth Tables (Example)
WEX 3-2-1: Construct a truth table for the logic statement: ∼ (P ∧ ∼ Q)
P Q
T
T
F
F
Josh Engwer (TTU)
∼Q P∧∼Q
∼ (P ∧ ∼ Q)
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
7 / 59
Truth Tables (Example)
WEX 3-2-1: Construct a truth table for the logic statement: ∼ (P ∧ ∼ Q)
P Q
T T
T F
F T
F F
Josh Engwer (TTU)
∼Q P∧∼Q
∼ (P ∧ ∼ Q)
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
8 / 59
Truth Tables (Example)
WEX 3-2-1: Construct a truth table for the logic statement: ∼ (P ∧ ∼ Q)
P Q
T T
T F
F T
F F
Josh Engwer (TTU)
∼Q P∧∼Q
F
T
F
T
∼ (P ∧ ∼ Q)
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
9 / 59
Truth Tables (Example)
WEX 3-2-1: Construct a truth table for the logic statement: ∼ (P ∧ ∼ Q)
P Q
T T
T F
F T
F F
Josh Engwer (TTU)
∼Q P∧∼Q
F
F
T
T
F
F
T
F
∼ (P ∧ ∼ Q)
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
10 / 59
Truth Tables (Example)
WEX 3-2-1: Construct a truth table for the logic statement: ∼ (P ∧ ∼ Q)
P Q
T T
T F
F T
F F
Josh Engwer (TTU)
∼Q P∧∼Q
F
F
T
T
F
F
T
F
∼ (P ∧ ∼ Q)
T
F
T
T
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
11 / 59
Truth Tables (Example)
WEX 3-2-1: Construct a truth table for the logic statement: ∼ (P ∧ ∼ Q)
P Q
T T
T F
F T
F F
Josh Engwer (TTU)
∼Q P∧∼Q
F
F
T
T
F
F
T
F
∼ (P ∧ ∼ Q)
T
F
T
T
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
12 / 59
Truth Tables (Example)
WEX 3-2-1: Construct a truth table for the logic statement: ∼ (P ∧ ∼ Q)
P Q
T T
T F
F T
F F
∼Q P∧∼Q
F
F
T
T
F
F
T
F
∼ (P ∧ ∼ Q)
T
F
T
T
While not required, let’s do a quick a posteriori analysis of the truth table:
If P is true and Q is true, then ∼ (P ∧ ∼ Q) is true.
Josh Engwer (TTU)
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
13 / 59
Truth Tables (Example)
WEX 3-2-1: Construct a truth table for the logic statement: ∼ (P ∧ ∼ Q)
P Q
T T
T F
F T
F F
∼Q P∧∼Q
F
F
T
T
F
F
T
F
∼ (P ∧ ∼ Q)
T
F
T
T
While not required, let’s do a quick a posteriori analysis of the truth table:
If P is true and Q is false, then ∼ (P ∧ ∼ Q) is false.
Josh Engwer (TTU)
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
14 / 59
Truth Tables (Example)
WEX 3-2-1: Construct a truth table for the logic statement: ∼ (P ∧ ∼ Q)
P Q
T T
T F
F T
F F
∼Q P∧∼Q
F
F
T
T
F
F
T
F
∼ (P ∧ ∼ Q)
T
F
T
T
While not required, let’s do a quick a posteriori analysis of the truth table:
If P is false and Q is true, then ∼ (P ∧ ∼ Q) is true.
Josh Engwer (TTU)
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
15 / 59
Truth Tables (Example)
WEX 3-2-1: Construct a truth table for the logic statement: ∼ (P ∧ ∼ Q)
P Q
T T
T F
F T
F F
∼Q P∧∼Q
F
F
T
T
F
F
T
F
∼ (P ∧ ∼ Q)
T
F
T
T
While not required, let’s do a quick a posteriori analysis of the truth table:
If P is false and Q is false, then ∼ (P ∧ ∼ Q) is true.
Josh Engwer (TTU)
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
16 / 59
Truth Tables (Example)
WEX 3-2-2: Construct a truth table for the logic statement: (P ∧ Q) ∧ ∼ R
Josh Engwer (TTU)
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
17 / 59
Truth Tables (Example)
WEX 3-2-2: Construct a truth table for the logic statement: (P ∧ Q) ∧ ∼ R
P Q
Josh Engwer (TTU)
R
(P ∧ Q) ∼ R
(P ∧ Q) ∧ ∼ R
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
18 / 59
Truth Tables (Example)
WEX 3-2-2: Construct a truth table for the logic statement: (P ∧ Q) ∧ ∼ R
P Q
T
T
T
T
F
F
F
F
Josh Engwer (TTU)
R
(P ∧ Q) ∼ R
(P ∧ Q) ∧ ∼ R
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
19 / 59
Truth Tables (Example)
WEX 3-2-2: Construct a truth table for the logic statement: (P ∧ Q) ∧ ∼ R
P Q
T T
T T
T F
T F
F T
F T
F F
F F
Josh Engwer (TTU)
R
(P ∧ Q) ∼ R
(P ∧ Q) ∧ ∼ R
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
20 / 59
Truth Tables (Example)
WEX 3-2-2: Construct a truth table for the logic statement: (P ∧ Q) ∧ ∼ R
P Q
T T
T T
T F
T F
F T
F T
F F
F F
Josh Engwer (TTU)
R
T
F
T
F
T
F
T
F
(P ∧ Q) ∼ R
(P ∧ Q) ∧ ∼ R
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
21 / 59
Truth Tables (Example)
WEX 3-2-2: Construct a truth table for the logic statement: (P ∧ Q) ∧ ∼ R
P Q
T T
T T
T F
T F
F T
F T
F F
F F
Josh Engwer (TTU)
R
T
F
T
F
T
F
T
F
(P ∧ Q) ∼ R
T
T
F
F
F
F
F
F
(P ∧ Q) ∧ ∼ R
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
22 / 59
Truth Tables (Example)
WEX 3-2-2: Construct a truth table for the logic statement: (P ∧ Q) ∧ ∼ R
P Q
T T
T T
T F
T F
F T
F T
F F
F F
Josh Engwer (TTU)
R
T
F
T
F
T
F
T
F
(P ∧ Q) ∼ R
T
F
T
T
F
F
F
T
F
F
F
T
F
F
F
T
(P ∧ Q) ∧ ∼ R
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
23 / 59
Truth Tables (Example)
WEX 3-2-2: Construct a truth table for the logic statement: (P ∧ Q) ∧ ∼ R
P Q
T T
T T
T F
T F
F T
F T
F F
F F
Josh Engwer (TTU)
R
T
F
T
F
T
F
T
F
(P ∧ Q) ∼ R
T
F
T
T
F
F
F
T
F
F
F
T
F
F
F
T
(P ∧ Q) ∧ ∼ R
F
T
F
F
F
F
F
F
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
24 / 59
Truth Tables (Example)
WEX 3-2-2: Construct a truth table for the logic statement: (P ∧ Q) ∧ ∼ R
P Q
T T
T T
T F
T F
F T
F T
F F
F F
Josh Engwer (TTU)
R
T
F
T
F
T
F
T
F
(P ∧ Q) ∼ R
T
F
T
T
F
F
F
T
F
F
F
T
F
F
F
T
(P ∧ Q) ∧ ∼ R
F
T
F
F
F
F
F
F
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
25 / 59
Truth Tables (Example)
WEX 3-2-2: Construct a truth table for the logic statement: (P ∧ Q) ∧ ∼ R
P Q
T T
T T
T F
T F
F T
F T
F F
F F
R
T
F
T
F
T
F
T
F
(P ∧ Q) ∼ R
T
F
T
T
F
F
F
T
F
F
F
T
F
F
F
T
(P ∧ Q) ∧ ∼ R
F
T
F
F
F
F
F
F
While not required, let’s do a quick a posteriori analysis of the truth table:
If P is true, Q is true, and R is true, then (P ∧ Q) ∧ ∼ R is false.
Josh Engwer (TTU)
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
26 / 59
Truth Tables (Example)
WEX 3-2-2: Construct a truth table for the logic statement: (P ∧ Q) ∧ ∼ R
P Q
T T
T T
T F
T F
F T
F T
F F
F F
R
T
F
T
F
T
F
T
F
(P ∧ Q) ∼ R
T
F
T
T
F
F
F
T
F
F
F
T
F
F
F
T
(P ∧ Q) ∧ ∼ R
F
T
F
F
F
F
F
F
While not required, let’s do a quick a posteriori analysis of the truth table:
If P is true, Q is true, and R is false, then (P ∧ Q) ∧ ∼ R is true.
Josh Engwer (TTU)
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
27 / 59
Truth Tables (Example)
WEX 3-2-2: Construct a truth table for the logic statement: (P ∧ Q) ∧ ∼ R
P Q
T T
T T
T F
T F
F T
F T
F F
F F
R
T
F
T
F
T
F
T
F
(P ∧ Q) ∼ R
T
F
T
T
F
F
F
T
F
F
F
T
F
F
F
T
(P ∧ Q) ∧ ∼ R
F
T
F
F
F
F
F
F
While not required, let’s do a quick a posteriori analysis of the truth table:
If P is false, Q is true, and R is false, then (P ∧ Q) ∧ ∼ R is false.
Josh Engwer (TTU)
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
28 / 59
Logical Equivalence (Definition)
Definition
(Logical Equivalence)
Two logic statements are logically equivalent, if they have the same
variables (e.g. P, Q, R, . . . ) and the final columns in their truth tables are
identical.
NOTATION: The symbol ⇐⇒ means ”is logically equivalent to”
Josh Engwer (TTU)
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
29 / 59
Logical Equivalence (Example)
WEX 3-2-3: Determine whether P ∧ (Q ∨ ∼ R) ⇐⇒ P ∧ ∼ (∼ Q ∧ R) or not.
Josh Engwer (TTU)
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
30 / 59
Logical Equivalence (Example)
WEX 3-2-3: Determine whether P ∧ (Q ∨ ∼ R) ⇐⇒ P ∧ ∼ (∼ Q ∧ R) or not.
P
Josh Engwer (TTU)
Q
R
∼R
Q∨∼R
P ∧ (Q ∨ ∼ R)
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
31 / 59
Logical Equivalence (Example)
WEX 3-2-3: Determine whether P ∧ (Q ∨ ∼ R) ⇐⇒ P ∧ ∼ (∼ Q ∧ R) or not.
P
T
T
T
T
F
F
F
F
Josh Engwer (TTU)
Q
R
∼R
Q∨∼R
P ∧ (Q ∨ ∼ R)
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
32 / 59
Logical Equivalence (Example)
WEX 3-2-3: Determine whether P ∧ (Q ∨ ∼ R) ⇐⇒ P ∧ ∼ (∼ Q ∧ R) or not.
P
T
T
T
T
F
F
F
F
Josh Engwer (TTU)
Q
T
T
F
F
T
T
F
F
R
∼R
Q∨∼R
P ∧ (Q ∨ ∼ R)
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
33 / 59
Logical Equivalence (Example)
WEX 3-2-3: Determine whether P ∧ (Q ∨ ∼ R) ⇐⇒ P ∧ ∼ (∼ Q ∧ R) or not.
P
T
T
T
T
F
F
F
F
Josh Engwer (TTU)
Q
T
T
F
F
T
T
F
F
R
T
F
T
F
T
F
T
F
∼R
Q∨∼R
P ∧ (Q ∨ ∼ R)
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
34 / 59
Logical Equivalence (Example)
WEX 3-2-3: Determine whether P ∧ (Q ∨ ∼ R) ⇐⇒ P ∧ ∼ (∼ Q ∧ R) or not.
P
T
T
T
T
F
F
F
F
Josh Engwer (TTU)
Q
T
T
F
F
T
T
F
F
R
T
F
T
F
T
F
T
F
∼R
F
T
F
T
F
T
F
T
Q∨∼R
P ∧ (Q ∨ ∼ R)
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
35 / 59
Logical Equivalence (Example)
WEX 3-2-3: Determine whether P ∧ (Q ∨ ∼ R) ⇐⇒ P ∧ ∼ (∼ Q ∧ R) or not.
P
T
T
T
T
F
F
F
F
Josh Engwer (TTU)
Q
T
T
F
F
T
T
F
F
R
T
F
T
F
T
F
T
F
∼R
F
T
F
T
F
T
F
T
Q∨∼R
T
T
F
T
T
T
F
T
P ∧ (Q ∨ ∼ R)
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
36 / 59
Logical Equivalence (Example)
WEX 3-2-3: Determine whether P ∧ (Q ∨ ∼ R) ⇐⇒ P ∧ ∼ (∼ Q ∧ R) or not.
P
T
T
T
T
F
F
F
F
Josh Engwer (TTU)
Q
T
T
F
F
T
T
F
F
R
T
F
T
F
T
F
T
F
∼R
F
T
F
T
F
T
F
T
Q∨∼R
T
T
F
T
T
T
F
T
P ∧ (Q ∨ ∼ R)
T
T
F
T
F
F
F
F
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
37 / 59
Logical Equivalence (Example)
WEX 3-2-3: Determine whether P ∧ (Q ∨ ∼ R) ⇐⇒ P ∧ ∼ (∼ Q ∧ R) or not.
P
Q
Josh Engwer (TTU)
P Q R ∼ R Q ∨ ∼ R P ∧ (Q ∨ ∼ R)
T T T
F
T
T
T
T
T
T T F
T F T
F
F
F
T
T
T
T F F
F T T
F
T
F
F T F
T
T
F
F F T
F
F
F
F F F
T
T
F
R ∼ Q ∼ Q ∧ R ∼ (∼ Q ∧ R) P ∧ ∼ (∼ Q ∧ R)
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
38 / 59
Logical Equivalence (Example)
WEX 3-2-3: Determine whether P ∧ (Q ∨ ∼ R) ⇐⇒ P ∧ ∼ (∼ Q ∧ R) or not.
P
T
T
T
T
F
F
F
F
Q
T
T
F
F
T
T
F
F
Josh Engwer (TTU)
P Q R ∼ R Q ∨ ∼ R P ∧ (Q ∨ ∼ R)
T T T
F
T
T
T
T
T
T T F
T F T
F
F
F
T
T
T
T F F
F T T
F
T
F
F T F
T
T
F
F F T
F
F
F
F F F
T
T
F
R ∼ Q ∼ Q ∧ R ∼ (∼ Q ∧ R) P ∧ ∼ (∼ Q ∧ R)
T
F
T
F
T
F
T
F
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
39 / 59
Logical Equivalence (Example)
WEX 3-2-3: Determine whether P ∧ (Q ∨ ∼ R) ⇐⇒ P ∧ ∼ (∼ Q ∧ R) or not.
P
T
T
T
T
F
F
F
F
Q
T
T
F
F
T
T
F
F
Josh Engwer (TTU)
P Q R ∼ R Q ∨ ∼ R P ∧ (Q ∨ ∼ R)
T T T
F
T
T
T
T
T
T T F
T F T
F
F
F
T
T
T
T F F
F T T
F
T
F
F T F
T
T
F
F F T
F
F
F
F F F
T
T
F
R ∼ Q ∼ Q ∧ R ∼ (∼ Q ∧ R) P ∧ ∼ (∼ Q ∧ R)
T
F
F
F
T
T
F
T
T
F
F
F
T
T
F
T
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
40 / 59
Logical Equivalence (Example)
WEX 3-2-3: Determine whether P ∧ (Q ∨ ∼ R) ⇐⇒ P ∧ ∼ (∼ Q ∧ R) or not.
P
T
T
T
T
F
F
F
F
Q
T
T
F
F
T
T
F
F
Josh Engwer (TTU)
P Q R ∼ R Q ∨ ∼ R P ∧ (Q ∨ ∼ R)
T T T
F
T
T
T
T
T
T T F
T F T
F
F
F
T
T
T
T F F
F T T
F
T
F
F T F
T
T
F
F F T
F
F
F
F F F
T
T
F
R ∼ Q ∼ Q ∧ R ∼ (∼ Q ∧ R) P ∧ ∼ (∼ Q ∧ R)
T
F
F
F
F
F
T
T
T
F
T
F
T
F
F
F
F
F
T
T
T
F
T
F
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
41 / 59
Logical Equivalence (Example)
WEX 3-2-3: Determine whether P ∧ (Q ∨ ∼ R) ⇐⇒ P ∧ ∼ (∼ Q ∧ R) or not.
P
T
T
T
T
F
F
F
F
Q
T
T
F
F
T
T
F
F
Josh Engwer (TTU)
P Q R ∼ R Q ∨ ∼ R P ∧ (Q ∨ ∼ R)
T T T
F
T
T
T
T
T
T T F
T F T
F
F
F
T
T
T
T F F
F T T
F
T
F
F T F
T
T
F
F F T
F
F
F
F F F
T
T
F
R ∼ Q ∼ Q ∧ R ∼ (∼ Q ∧ R) P ∧ ∼ (∼ Q ∧ R)
T
F
F
T
F
F
F
T
T
T
T
F
F
T
F
T
T
F
F
T
F
F
F
T
T
T
T
F
F
T
F
T
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
42 / 59
Logical Equivalence (Example)
WEX 3-2-3: Determine whether P ∧ (Q ∨ ∼ R) ⇐⇒ P ∧ ∼ (∼ Q ∧ R) or not.
P
T
T
T
T
F
F
F
F
Q
T
T
F
F
T
T
F
F
Josh Engwer (TTU)
P Q R ∼ R Q ∨ ∼ R P ∧ (Q ∨ ∼ R)
T T T
F
T
T
T
T
T
T T F
T F T
F
F
F
T
T
T
T F F
F T T
F
T
F
F T F
T
T
F
F F T
F
F
F
F F F
T
T
F
R ∼ Q ∼ Q ∧ R ∼ (∼ Q ∧ R) P ∧ ∼ (∼ Q ∧ R)
T
F
F
T
T
F
F
F
T
T
T
T
T
F
F
F
T
F
T
T
T
F
F
T
F
F
F
F
T
F
T
T
T
F
F
F
T
F
T
F
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
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Logical Equivalence (Example)
WEX 3-2-3: Determine whether P ∧ (Q ∨ ∼ R) ⇐⇒ P ∧ ∼ (∼ Q ∧ R) or not.
P
T
T
T
T
F
F
F
F
Q
T
T
F
F
T
T
F
F
Josh Engwer (TTU)
P Q R ∼ R Q ∨ ∼ R P ∧ (Q ∨ ∼ R)
T T T
F
T
T
T
T
T
T T F
T F T
F
F
F
T
T
T
T F F
F T T
F
T
F
F T F
T
T
F
F F T
F
F
F
F F F
T
T
F
R ∼ Q ∼ Q ∧ R ∼ (∼ Q ∧ R) P ∧ ∼ (∼ Q ∧ R)
T
F
F
T
T
F
F
F
T
T
T
T
T
F
F
F
T
F
T
T
T
F
F
T
F
F
F
F
T
F
T
T
T
F
F
F
T
F
T
F
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
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Logical Equivalence (Example)
WEX 3-2-3: Determine whether P ∧ (Q ∨ ∼ R) ⇐⇒ P ∧ ∼ (∼ Q ∧ R) or not.
P Q
T T
T T
T F
T F
F T
F T
F F
F F
R
T
F
T
F
T
F
T
F
∼R
F
T
F
T
F
T
F
T
Q∨∼R
T
T
F
T
T
T
F
T
P ∧ (Q ∨ ∼ R)
T
T
F
T
F
F
F
F
P ∧ ∼ (∼ Q ∧ R)
T
T
F
T
F
F
F
F
Since the corresponding entries (in blue) of
P ∧ (Q ∨ ∼ R) and P ∧ ∼ (∼ Q ∧ R) all match,
P ∧ (Q ∨ ∼ R) and P ∧ ∼ (∼ Q ∧ R) are logically equivalent.
Josh Engwer (TTU)
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
45 / 59
Simplifying Logic Statements (DeMorgan’s Laws)
Theorem
(DeMorgan’s Laws)
(a) ∼ (∼ P)
⇐⇒ P
(b) ∼ (P ∧ Q) ⇐⇒ (∼ P) ∨ (∼ Q)
(c) ∼ (P ∨ Q) ⇐⇒ (∼ P) ∧ (∼ Q)
Josh Engwer (TTU)
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
46 / 59
Simplifying Logic Statements (DeMorgan’s Laws)
Theorem
(DeMorgan’s Laws)
(a) ∼ (∼ P)
⇐⇒ P
(b) ∼ (P ∧ Q) ⇐⇒ (∼ P) ∨ (∼ Q)
(c) ∼ (P ∨ Q) ⇐⇒ (∼ P) ∧ (∼ Q)
PROOF:
(a)
Josh Engwer (TTU)
P
T
F
∼P
F
T
∼ (∼ P)
T
F
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
47 / 59
Simplifying Logic Statements (DeMorgan’s Laws)
Theorem
(DeMorgan’s Laws)
(a) ∼ (∼ P)
⇐⇒ P
(b) ∼ (P ∧ Q) ⇐⇒ (∼ P) ∨ (∼ Q)
(c) ∼ (P ∨ Q) ⇐⇒ (∼ P) ∧ (∼ Q)
PROOF:
(b)
P
T
T
F
F
Josh Engwer (TTU)
Q
T
F
T
F
P∧Q
T
F
F
F
∼ (P ∧ Q)
F
T
T
T
∼P
F
F
T
T
∼Q
F
T
F
T
Logic: Truth Tables, DeMorgan’s Laws
(∼ P) ∨ (∼ Q)
F
T
T
T
20 July 2015
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Simplifying Logic Statements (DeMorgan’s Laws)
Theorem
(DeMorgan’s Laws)
(a) ∼ (∼ P)
⇐⇒ P
(b) ∼ (P ∧ Q) ⇐⇒ (∼ P) ∨ (∼ Q)
(c) ∼ (P ∨ Q) ⇐⇒ (∼ P) ∧ (∼ Q)
PROOF:
(c)
P
T
T
F
F
Josh Engwer (TTU)
Q
T
F
T
F
P∨Q
T
T
T
F
∼ (P ∨ Q)
F
F
F
T
∼P
F
F
T
T
∼Q
F
T
F
T
Logic: Truth Tables, DeMorgan’s Laws
(∼ P) ∧ (∼ Q)
F
F
F
T
20 July 2015
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Simplifying Logic Statements (Example)
WEX 3-2-4: Via DeMorgan’s Laws, completely simplify: ∼ [P ∨ (∼ Q ∧ ∼ R)]
Josh Engwer (TTU)
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
50 / 59
Simplifying Logic Statements (Example)
WEX 3-2-4: Via DeMorgan’s Laws, completely simplify: ∼ [P ∨ (∼ Q ∧ ∼ R)]
∼ [P ∨ (∼ Q ∧ ∼ R)]
Josh Engwer (TTU)
⇐⇒
(∼ P) ∧ ∼ (∼ Q ∧ ∼ R)
Logic: Truth Tables, DeMorgan’s Laws
[DeMorgan (c)]
20 July 2015
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Simplifying Logic Statements (Example)
WEX 3-2-4: Via DeMorgan’s Laws, completely simplify: ∼ [P ∨ (∼ Q ∧ ∼ R)]
∼ [P ∨ (∼ Q ∧ ∼ R)]
Josh Engwer (TTU)
⇐⇒
⇐⇒
(∼ P) ∧ ∼ (∼ Q ∧ ∼ R)
[DeMorgan (c)]
(∼ P) ∧ ∼ (∼ Q) ∨ ∼ (∼ R) [DeMorgan (b)]
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
52 / 59
Simplifying Logic Statements (Example)
WEX 3-2-4: Via DeMorgan’s Laws, completely simplify: ∼ [P ∨ (∼ Q ∧ ∼ R)]
∼ [P ∨ (∼ Q ∧ ∼ R)]
Josh Engwer (TTU)
⇐⇒
⇐⇒
⇐⇒
(∼ P) ∧ ∼ (∼ Q ∧ ∼ R)
[DeMorgan (c)]
(∼ P) ∧ ∼ (∼ Q) ∨ ∼ (∼ R) [DeMorgan (b)]
(∼ P) ∧ Q ∨ R
[DeMorgan (a)]
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
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Negating English Statements (Example)
WEX 3-2-5: Via DeMorgan’s Laws, negate the English statement:
”Flowers are not red or John’s truck has anti-lock brakes.”
Josh Engwer (TTU)
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
54 / 59
Negating English Statements (Example)
WEX 3-2-5: Via DeMorgan’s Laws, negate the English statement:
”Flowers are not red or John’s truck has anti-lock brakes.”
Let P ≡ ”Flowers are not red”,
Josh Engwer (TTU)
Q ≡ ”John’s truck has anti-lock brakes.”
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
55 / 59
Negating English Statements (Example)
WEX 3-2-5: Via DeMorgan’s Laws, negate the English statement:
”Flowers are not red or John’s truck has anti-lock brakes.”
Let P ≡ ”Flowers are not red”,
Q ≡ ”John’s truck has anti-lock brakes.”
Then, ”Flowers are not red or John’s truck has anti-lock brakes.” ≡ P ∨ Q
Josh Engwer (TTU)
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
56 / 59
Negating English Statements (Example)
WEX 3-2-5: Via DeMorgan’s Laws, negate the English statement:
”Flowers are not red or John’s truck has anti-lock brakes.”
Let P ≡ ”Flowers are not red”,
Q ≡ ”John’s truck has anti-lock brakes.”
Then, ”Flowers are not red or John’s truck has anti-lock brakes.” ≡ P ∨ Q
Negation: ∼ (P ∨ Q) ⇐⇒ (∼ P) ∧ (∼ Q)
Josh Engwer (TTU)
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
57 / 59
Negating English Statements (Example)
WEX 3-2-5: Via DeMorgan’s Laws, negate the English statement:
”Flowers are not red or John’s truck has anti-lock brakes.”
Let P ≡ ”Flowers are not red”,
Q ≡ ”John’s truck has anti-lock brakes.”
Then, ”Flowers are not red or John’s truck has anti-lock brakes.” ≡ P ∨ Q
Negation: ∼ (P ∨ Q) ⇐⇒ (∼ P) ∧ (∼ Q)
⇐⇒ ”Flowers are red and John’s truck does not have anti-lock brakes.”
Josh Engwer (TTU)
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
58 / 59
Fin
Fin.
Josh Engwer (TTU)
Logic: Truth Tables, DeMorgan’s Laws
20 July 2015
59 / 59