Proofs Homework
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chapter1p1.pptxThe Foundations: Logic and Proofs
Chapter 1, Part I: Propositional Logic
Propositional Logic
Section 1.1
Propositions
A proposition is a declarative sentence that is either true or false.
Examples of propositions:
The Moon is made of green cheese.
Trenton is the capital of New Jersey.
Toronto is the capital of Canada.
1 + 0 = 1
0 + 0 = 2
Examples that are not propositions.
Sit down!
What time is it?
x + 1 = 2
x + y = z
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Propositional Logic
Constructing Propositions
Propositional Variables: p, q, r, s, …
The proposition that is always true is denoted by T and the proposition that is always false is denoted by F.
Compound Propositions; constructed from logical connectives and other propositions
Negation ¬
Conjunction ∧
Disjunction ∨
Exclusive Or ⊕
Implication →
Biconditional ↔
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Compound Propositions: Negation
The negation of a proposition p is denoted by ¬p and has this truth table:
Example: If p denotes “The earth is round.”, then ¬p denotes “It is not the case that the earth is round,” or more simply “The earth is not round.”
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| p | ¬p |
| T | F |
| F | T |
Conjunction
The conjunction of propositions p and q is denoted by p ∧ q and has this truth table:
Example: If p denotes “I am at home.” and q denotes “It is raining.” then p ∧q denotes “I am at home and it is raining.”
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| p | q | p ∧ q |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
Disjunction
The disjunction of propositions p and q is denoted by p ∨q and has this truth table:
Example: If p denotes “I am at home.” and q denotes “It is raining.” then p ∨q denotes “I am at home or it is raining.”
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| p | q | p ∨q |
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
The Connective Or in English
In English “or” has two distinct meanings.
“Inclusive Or” − In the sentence “Students who have taken CS202 or Math120 may take this class,” we assume that students need to have taken one of the prerequisites, but may have taken both. This is the meaning of disjunction. For p ∨q to be true, either one or both of p and q must be true.
“Exclusive Or” − When reading the sentence “Soup or salad comes with this entrée,” we do not expect to be able to get both soup and salad. This is the meaning of Exclusive Or (Xor). In p ⊕ q , one of p and q must be true, but not both. The truth table for ⊕ is:
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| p | q | p ⊕ q |
| T | T | F |
| T | F | T |
| F | T | T |
| F | F | F |
Implication
If p and q are propositions, then p →q is a conditional statement or implication which is read as “if p, then q ” and has this truth table:
Example: If p denotes “I am at home.” and q denotes “It is raining.” then p →q denotes “If I am at home then it is raining.”
In p →q , p is the hypothesis (antecedent or premise) and q is the conclusion (or consequent).
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| p | q | p →q |
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Implication
Hence p →q is often called a material conditional or material implication; it is truth functional; consider the alternatives:
The material conditional is adequate for mathematical reasoning; so, “If 1 + 1 = 0, then 1 = 2” is often called trivially true.
There are other logics (such as relevance logics) that reject the material conditional as the meaning of “if p, then q”, BUT these logics are not truth functional.
| p | q | p →q | Altern. 1 | Altern. 2 | Altern. 3 |
| T | T | T | T | T | T |
| T | F | F | F | F | F |
| F | T | T | T | F | F |
| F | F | T | F | T | F |
Understanding Implication
In p →q there does not need to be any connection between the antecedent or the consequent. The “meaning” of p →q depends only on the truth values of p and q.
These implications are true, but would not be used in ordinary English.
“If the moon is made of green cheese, then I have more money than Bill Gates. ”
“If the moon is made of green cheese then 1 + 1 = 2.”
“If 1 + 1 = 3, then your grandma wears combat boots.”
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Different Ways of Expressing p →q
if p, then q p implies q
if p, q p only if q
q unless ¬p ¬p or q
q, if p q when p
q whenever p p is sufficient for q
q follows from p q is necessary for p
a necessary condition for p is q
a sufficient condition for q is p
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Converse, Contrapositive, and Inverse
From p →q we can form new conditional statements .
q →p is the converse of p →q
¬q → ¬ p is the contrapositive of p →q
¬ p → ¬ q is the inverse of p →q
Example: Find the converse, inverse, and contrapositive of “It raining is a sufficient condition for my not going to town,” or “If it is raining, I will not go to town.”
Solution (note the simplification of double negations):
converse: If I do not go to town, then it is raining.
inverse: If it is not raining, then I will go to town.
contrapositive: If I go to town, then it is not raining.
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Biconditional
If p and q are propositions, then we can form the biconditional proposition p ↔q , read as “p if and only if q .” The biconditional: p ↔q denotes the proposition with this truth table:
If p denotes “I am at home.” and q denotes “It is raining.” then p ↔q denotes “I am at home if and only if it is raining.”
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| p | q | p ↔q |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
Expressing the Biconditional
Some alternative ways “p if and only if q” is expressed in English:
p is necessary and sufficient for q
if p then q , and conversely
p just in case q
p iff q
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Truth Tables For Compound Propositions
Construction of a truth table:
Rows
Need a row for every possible combination of truth values for the atomic propositions.
Columns
Need a column for the compound proposition (usually at far right)
Need a column for the truth value of each expression that occurs in the compound proposition as it is built up.
This includes the atomic propositions
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Example Truth Table
Construct a truth table for
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| p | q | r | r | p q | p q → r |
| T | T | T | F | T | F |
| T | T | F | T | T | T |
| T | F | T | F | T | F |
| T | F | F | T | T | T |
| F | T | T | F | T | F |
| F | T | F | T | T | T |
| F | F | T | F | F | T |
| F | F | F | T | F | T |
Equivalent Propositions
Two propositions are equivalent if they always have the same truth value.
Example: Show using a truth table that the conditional is equivalent to the contrapositive.
Solution:
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| p | q | ¬ p | ¬ q | p →q | ¬q → ¬ p |
| T | T | F | F | T | T |
| T | F | F | T | F | F |
| F | T | T | F | T | T |
| F | F | T | T | T | T |
Using a Truth Table to Show Non-Equivalence
Example: Show using truth tables that neither the converse nor inverse of an implication are equivalent to the implication.
Solution:
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| p | q | ¬ p | ¬ q | p →q | ¬ p →¬ q | q → p |
| T | T | F | F | T | T | T |
| T | F | F | T | F | T | T |
| F | T | T | F | T | F | F |
| F | F | T | T | T | T | T |
Problem
How many rows are there in a truth table with n propositional variables?
Solution: 2n
Note that this means that with n propositional variables, we can construct 2n distinct (i.e., not equivalent) assignments of truth value to the variables.
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Precedence of Logical Operators
| Operator | Precedence |
| | 1 |
| | 2 3 |
| | 4 5 |
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p q r is equivalent to (p q) r
If the intended meaning is p (q r )
then parentheses must be used.
In-Class Exercises
§1.1: #s 10a-d, 14a-d, 18, 22b-e, & 32a-d
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Applications of Propositional Logic
Section 1.2
Translating English Sentences
Steps to convert an English sentence to a statement in propositional logic
Identify atomic propositions and represent using propositional constants.
Determine appropriate logical connectives
“If I go to Harry’s or to the country, I will not go shopping.”
p: I go to Harry’s
q: I go to the country.
r: I will go shopping.
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If p or q then not r.
Example
Problem: Translate the following sentence into propositional logic:
“You can access the Internet from campus only if you are a computer science major or you are not a freshman.”
One Solution: Let a, c, and f represent respectively “You can access the internet from campus,” “You are a computer science major,” and “You are a freshman.”
a→ (c ∨ ¬ f )
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System Specifications
System and Software engineers take requirements in English and express them in a precise specification language based on logic.
Example: Express in propositional logic:
“The automated reply cannot be sent when the file system is full”
Solution: One possible solution: Let p denote “The automated reply can be sent” and q denote “The file system is full.”
q→ ¬ p
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Consistent System Specifications
Definition: A list (or set) of propositions is consistent if it is possible to assign truth values to the proposition variables so that each proposition is true.
Exercise: Are these specifications consistent?
“The diagnostic message is stored in the buffer or it is retransmitted.”
“The diagnostic message is not stored in the buffer.”
“If the diagnostic message is stored in the buffer, then it is retransmitted.”
Solution: Let p denote “The diagnostic message is stored in the buffer.” Let q denote “The diagnostic message is retransmitted” The specification can be written as: p ∨ q, p→ q, ¬p. When p is false and q is true all three statements are true. So the specification is consistent.
What if “The diagnostic message is not retransmitted is added.”
Solution: Now we are adding ¬q and there is no satisfying assignment. So the specification is not consistent.
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Other Important Translations
p unless q: p ∨ q
neither p nor q: ¬ (p ∨ q) ¬ p ¬q
p only if q: p → q
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Logic Puzzles
An island has two kinds of inhabitants, knights, who always tell the truth, and knaves, who always lie.
You go to the island and meet A and B.
A says “B is a knight.”
B says “The two of us are of opposite types.”
Example: What are the types of A and B?
Solution: Let p and q be the statements that A is a knight and B is a knight, respectively. So, then p represents the proposition that A is a knave and q that B is a knave.
If A is a knight, then p is true. Since knights tell the truth, q must also be true. Then, as B asserts that (p ∧ q)∨ ( p ∧ q), it would have to be true, but it cannot be. So, A is not a knight and therefore p must be true.
If A is a knave and always lie, then B must not be a knight, and B is a knave. So, then both p and q hold since both are knaves.
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Raymond Smullyan
(Born 1919)
Electronic Circuits
Electronic circuits; each input/output signal can be viewed as a 0 or 1.
0 represents False
1 represents True
Complicated circuits are constructed from three basic circuits called gates.
The inverter (NOT gate)takes an input bit and produces the negation of that bit.
The OR gate takes two input bits and produces the value equivalent to the disjunction of the two bits.
The AND gate takes two input bits and produces the value equivalent to the conjunction of the two bits.
More complicated digital circuits can be constructed by combining these basic circuits to produce the desired output given the input signals by building a circuit for each piece of the output expression and then combining them. For example:
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In-Class Exercises
§1.2: #s 4, 12, & 40
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Propositional Equivalences
Section 1.3
Tautologies, Contradictions, and Contingencies
A tautology is a proposition which is always true.
Example: p ∨¬p
A contradiction is a proposition which is always false.
Example: p ∧¬p
A contingency is a proposition which is neither a tautology nor a contradiction, such as p
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| P | ¬p | p ∨¬p | p ∧¬p |
| T | F | T | F |
| F | T | T | F |
Logically Equivalence
Two compound propositions p and q are logically equivalent if p ↔ q is a tautology.
We write this as p ≡ q where p and q are compound propositions.
Two compound propositions p and q are equivalent if and only if the columns in a truth table giving their truth values agree.
This truth table shows ¬p ∨ q is equivalent to p → q.
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| p | q | ¬p | ¬p ∨ q | p→ q |
| T | T | F | T | T |
| T | F | F | F | F |
| F | T | T | T | T |
| F | F | T | T | T |
De Morgan’s Laws
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| p | q | ¬p | ¬q | (p ∨ q) | ¬(p ∨ q) | ¬p ∧ ¬q |
| T | T | F | F | T | F | F |
| T | F | F | T | T | F | F |
| F | T | T | F | T | F | F |
| F | F | T | T | F | T | T |
This truth table shows that De Morgan’s Second Law holds.
Augustus De Morgan
1806-1871
Key Logical Equivalences
Identity Laws: ,
Domination Laws: ,
Idempotent laws: ,
Double Negation Law:
Negation Laws: ,
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Key Logical Equivalences (cont)
Commutative Laws:
Associative Laws:
Distributive Laws:
Absorption Laws:
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More Logical Equivalences
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Constructing New Logical Equivalences
We can show that two expressions are logically equivalent by developing a series of logically equivalent statements.
To prove that can we produce a series of equivalences beginning with A and ending with B, using th laws above.
Keep in mind that whenever a proposition (represented by a propositional variable) occurs in the equivalences listed earlier, it may be replaced by an arbitrarily complex compound proposition.
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Equivalence Proofs
Example: Show that
is logically equivalent to
Solution:
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Equivalence Proofs
Example: Show that
is a tautology.
Solution:
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Note: T v T should be T v (¬p v ¬ q)
Propositional Satisfiability
A compound proposition is satisfiable if there is an assignment of truth values to its variables that make it true. When no such assignments exist, the compound proposition is unsatisfiable.
A compound proposition is unsatisfiable if and only if it is a contradiction (cannot be true)
A compound proposition is unsatisfiable if and only if its negation is a tautology.
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Questions on Propositional Satisfiability
Example: Determine the satisfiability of the following compound propositions:
Solution: Satisfiable. Assign T to p, q, and r.
Solution: Satisfiable. Assign T to p and F to q.
Solution: Not satisfiable. Check each possible assignment of truth values to the propositional variables and none will make the proposition true.
In-Class Exercises
§1.3: #s 8a & b, 14, 22, 32, & 61a
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Disjunctive Normal Form (optional)
A propositional formula is in disjunctive normal form if it consists of a disjunction of (1, … ,n) disjuncts where each disjunct consists of a conjunction of (1, …, m) atomic formulas or the negation of an atomic formula.
Yes
No
Disjunctive Normal Form is important for the circuit design methods discussed in Chapter 12.
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Disjunctive Normal Form (optional)
Example: Show that every compound proposition can be put in disjunctive normal form.
Solution: Construct the truth table for the proposition. Then an equivalent proposition is the disjunction with n disjuncts (where n is the number of rows for which the formula evaluates to T). Each disjunct has m conjuncts where m is the number of distinct propositional variables. Each conjunct includes the positive form of the propositional variable if the variable is assigned T in that row and the negated form if the variable is assigned F in that row. This proposition is in disjunctive normal from.
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Disjunctive Normal Form (optional)
Example: Find the Disjunctive Normal Form (DNF) of
(p ∨ q) → ¬r
Solution: This proposition is true when r is false or when both p and q are false.
(¬ p∧ ¬ q) ∨ ¬r
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Conjunctive Normal Form (optional)
A compound proposition is in Conjunctive Normal Form (CNF) if it is a conjunction of disjunctions.
Every proposition can be put in an equivalent CNF.
Conjunctive Normal Form (CNF) can be obtained by eliminating implications, moving negation inwards and using the distributive and associative laws.
Important in resolution theorem proving used in artificial Intelligence (AI).
A compound proposition can be put in conjunctive normal form through repeated application of the logical equivalences covered earlier.
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Conjunctive Normal Form (optional)
Example: Put the following into CNF:
Solution:
Eliminate implication signs:
Move negation inwards; eliminate double negation:
Convert to CNF using associative/distributive laws
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Notation
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Needed for the next example.
Sudoku (optional)
A Sudoku puzzle is represented by a 99 grid made up of nine 33 subgrids, known as blocks. Some of the 81 cells of the puzzle are assigned one of the numbers 1,2, …, 9.
The puzzle is solved by assigning numbers to each blank cell so that every row, column and block contains each of the nine possible numbers.
Example
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Encoding as a Satisfiability Problem
Let p(i,j,n) denote the proposition that is true when the number n is in the cell in the ith row and the jth column.
There are 99 9 = 729 such propositions.
In the sample puzzle p(2,4,5) is true, but p(2,j,5) is false for j = 1,3,…,9
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Encoding (cont)
For each cell with a given value, assert p(i,j,n), when the cell in row i and column j has the given value.
Assert that every row contains every number.
Assert that every column contains every number.
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Encoding (cont)
Assert that each of the 3 x 3 blocks contain every number.
(this is tricky - ideas from chapter 4 help)
Assert that no cell contains more than one number. Take the conjunction over all values of n, n’, i, and j, where each variable ranges from 1 to 9 and
of
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Solving Satisfiability Problems
To solve a Sudoku puzzle, we need to find an assignment of truth values to the 729 variables of the form p(i,j,n) that makes the conjunction of the assertions true. Those variables that are assigned T yield a solution to the puzzle.
A truth table can always be used to determine the satisfiability of a compound proposition. But this is too complex even for modern computers for large problems.
There has been much work on developing efficient methods for solving satisfiability problems as many practical problems can be translated into satisfiability problems.
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