1. COMP 1380 Discrete Structures I Thompson Rivers University
Logic and Truth Tables
COMP 1380 Discrete Structures I
Computing Science
Thompson Rivers University

2. Course Contents Speaking Mathematically – .5 weeks
Number Systems and Computer Arithmetic – 2 weeks
Logic and Truth Tables – 1 week
Boolean Algebra and Logic Gates – 1 week
Vectors and Matrices – 2 weeks
Sets and Counting – 1.5 weeks
Probability Theory and Distributions – 2 weeks
Statics and Random Variables – 2 weeks
Not the replacement of IPv6; Inter-transition mechanism; But if IPv6 would fail
TRU-COMP1380
Logic and Truth Tables

3. Unit Learning Objectives
Recall truth tables for ~, , , , .
Give a truth table for a given composite statement.
Recall logical equivalences.
Simplify a compound statement using logical equivalences.
Infer a conclusion from a compound statement.
TRU-COMP1380
Logic and Truth Tables

4. Unit Contents Sections 2.1 – 2.3 from the textbook
Logical Form and Logical Equivalence
Conditional Statements
Valid and Invalid Arguments
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Logic and Truth Tables

5. Logical Form and Logical Equivalence
Argument = premises + conclusion
To have confidence in the conclusion in your argument, the premises should be acceptable on their own merits or follow from other statements that are known to be true.
Logical forms for valid arguments?
Examples
Argument 1: If the program syntax is faulty or if program execution results in division by zero, then the computer will generate an error message. Therefore, if the computer does not generate, then the program syntax is correct and program execution does not result in division by zero.
Argument 2: If x is a real number such that x < -2 or x > 2, then x2 > 4. Therefore, if x2 /> 4, then x /< -2 and x /> 2.
The common logical form of both of the above arguments:
If p or q, then r. Therefore, if not r, then not p and not q.
[Q] Is the above logical form valid?
TRU-COMP1380
Logic and Truth Tables

6. Definition of statement
A statement (or proposition) is a sentence that is true or false but not both.
Examples
Two plus two equals four.
2 + 2 = 4
I am a TRU student.
x + y > 0 ???
TRU-COMP1380
Logic and Truth Tables

7. Compound Statements Symbols used in complicated logical statements:
~ not ~p negation of p
 and p  q conjunction of p and q
 or p  q disjunction of p and q
 exclusive or p  q
Order of operations: ( ) and ~ have the precedence.
~p  q = (~p)  q
~(p  q)
TRU-COMP1380
Logic and Truth Tables

8. Example It is not hot but it is sunny. It is neither hot nor sunny.
-> It is not hot, and it is sunny. It is not hot, and it is not sunny.
Let h  “it is hot” and s  “it is sunny.” Then the above statements can be translated as
~h  s ~h  ~s
Suppose x is a particular real number. Let p, q, and r symbolize
“0 < x,” “x < 3,” and “x = 3.” respectively.
Then the following inequalities
x  3 0 < x < 3 0 < x  3
can be translated as
q  r p  q p  (q  r)
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Logic and Truth Tables

9. Truth Tables Negation Conjunction Disjunction p q p  q T ? F p q
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Logic and Truth Tables

10. Truth Tables Exclusive or p q p  q T F TRU-COMP1380
Logic and Truth Tables

11. Example ~p  q (p  q)  ~(p  q) (p  q)  ~r p q ~p  q T ? F
~T  T = ?
(p  q)  ~(p  q)
Can you write a truth table?
When p is T and q is F?
(p  q)  ~r
When p is T, q is F and r is F? p q
~p  q T ? F
TRU-COMP1380
Logic and Truth Tables

12. Logical Equivalence Example 6 > 2  2 < 6 How to prove
6 > 2  2 < 6
How to prove
p  q  q  p Commutative law
p  q  q  p Commutative law
~(~p)  p Double negative law
~(p  q) ≠ ~p  ~q ???
~(p  q) ≠ ~p  ~q ???
TRU-COMP1380
Logic and Truth Tables

13. De Morgan’s Laws Examples ~(p  q) = ~p  ~q Can you prove it?
~(statement1  statement2) = ~statement1  ~statement2
~(p  q) = ~p  ~q
~(statement1  statement2) = ~statement1  ~statement2
Examples
~(~p  q) = ??? ~(p  ~q) = ???
The negation of “John is 6 feet tall and he weighs at least 200 pounds.” is
...
The negation of “The bus was late or Tom’s watch was slow.” is
The negation of “-1 < x  4” is
TRU-COMP1380
Logic and Truth Tables

14. Tautologies and Contradictions
p  ~p = T; p  T = T Always true
p  ~p = F; p  F = F Always false
Some other interesting equivalences
p  F = p Can you prove it?
p  T = p
Associative Laws
(p  q)  r = p  (q  r)
(p  q)  r = p  (q  r)
Distributive Laws
p  (q  r) = (p  q)  (p  r) Can you prove it?
p  (q  r) = (p  q)  (p  r)
TRU-COMP1380
Logic and Truth Tables

15. Prove ~(~p  q)  (p  q) = p using equivalences.
~(~p  q)  (p  q) = (~(~p)  ~q)  (p  q)
= ...
(p  ~q)  p = ??? Can you simplify using equivalences?
~((~p  q)  (~p  ~q)) = ???
TRU-COMP1380
Logic and Truth Tables

16. Conditional Statements
If hypothesis (or antecedent), then conclusion (or consequent).
hypothesis  conclusion
Example
If 4686 is divisible by 6, then 4686 is divisible by 3. Is this statement True?
Let p = “4686 is divisible by 6,” and q = “4686 is divisible by 3”. Then
p  q
Truth table for p  q
p  q ≡ ~p  q ??? p q
p  q T F
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Logic and Truth Tables

17. p  q ≡ ~q  ~p ??? Contrapositive = ~(~(p  q))
p  q  r ≡ (p  r)  (q  r) ???
= ~(p  q)  r
= ...
~(p  q) ≡ p  ~q ???
= ~(~p  q)
p  q ≡ ~q  ~p ??? Contrapositive
= ~(~(p  q))
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Logic and Truth Tables

18. The biconditional of p and q p  q ≡ (p  q)  (q  p)
p if and only if q
p iff q
The truth table is p q
p  q T ? F
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Logic and Truth Tables

19. Valid and Invalid Arguments
Example
If (p  (q  ~r)) and (q  (p  r)), then is (p  r) valid?
What do we have to do?
p  r must be true for all the cases in which both of (p  (q  ~r)) and (q  (p  r)) are true. We may use the truth table to see if the statement is valid.
Modus Ponents
Both of p  q and p are valid, then q is valid.
(p  q )  p = (~p  q )  p = (F  q )  T = q, or by truth table
Therefore q must be T when (p  q ) and p are T.
If x is a human, then x is mortal. (It is like a rule.)
Dave is a human. (It is like observation.)
Therefore Dave is mortal. (It is like a conclusion.)
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Logic and Truth Tables

20. Modus Tollens
If p  q and ~q are valid, then ~p is valid.
How to prove?
If x is human, then x is mortal. (rule)
But Zeus is not mortal. (observation)
Therefore (conclusion)
If x is divisible by 6, the x is divisible by 3. (rule)
(Instantiation: …)
14 is not divisible by 3. (fact)
If a city is big, then the city has tall buildings. Because Kamloops have a tall building, Kamloops is a big city. Is this argument valid?
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Logic and Truth Tables

21. Generalization Specialization Elimination Transitivity
p, then p  q How to prove?
Specialization
p  q, then p How to prove?
Elimination
p  q and q, then p How to prove?
Transitivity
If p  q and q  r are valid, then p  r is valid. How to show?
Contradiction Rule
If p  F is valid, then ~p is valid. How to show?
If ~p  F is valid, then p is valid.
The logical heart of the method of proof by contradiction. If an assumption leads to a contradiction, then that assumption must be false. p q
p  q T F
TRU-COMP1380
Logic and Truth Tables