Download CS311H: Discrete Mathematics Inference Rules for Quantifiers

Review
Name
Modus ponens
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Modus tollens
Inference Rules for Quantifiers
Hypothetical syllogism
Or introduction
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Or elimination
And introduction
And elimination
Resolution
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Işıl Dillig,
Example
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1. It is not raining or Kate has her umbrella
¬φ1
φ1 → φ2
φ2 → φ3
φ1 → φ3
φ1
φ1 ∨ φ2
φ1 ∨ φ2
¬φ2
φ1
φ1
φ2
φ1 ∧ φ2
φ1 ∧ φ2
φ1
φ1 ∨ φ2
¬φ1 ∨ φ3
φ2 ∨ φ3
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First, encode hypotheses and conclusion as logical formulas.
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To do this, identify propositions used in the argument:
2. Kate does not have her umbrella or she does not get wet
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r = ”It is raining”
3. It is raining or Kate does not get wet
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u= ”Kate has her umbrella”
4. Kate is grumpy only if she is wet
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w = ”Kate is wet”
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g = ”Kate is grumpy”
Show these lead to the conclusion: ”Kate is not grumpy.”
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Işıl Dillig,
Encoding in Logic, cont.
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φ2
φ1 → φ2
¬φ2
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Encoding in Logic
Assume the following hypotheses:
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Rule of Inference
φ1
φ1 → φ2
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Formal Proof Using Inference Rules
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”It is not raining or Kate has her umbrella.”
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”Kate does not have her umbrella or she does not get wet”
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”It is raining or Kate does not get wet.”
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” Kate is grumpy only if she is wet.”
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Conclusion: ”Kate is not grumpy.”
CS311H: Discrete Mathematics
CS311H: Discrete Mathematics
Inference Rules for Quantifiers
1.
2.
3.
4.
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Işıl Dillig,
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¬r ∨ u
¬u ∨ ¬w
r ∨ ¬w
g →w
Hypothesis
Hypothesis
Hypothesis
Hypothesis
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Additional Inference Rules for Quantified Formulas
Universal Instantiation
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Inference rules we learned so far are sufficient for reasoning
about quantifier-free statements
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If we know something is true for all members of a group, we
can conclude it is also true for a specific member of this group
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Four more inference rules for making deductions from
quantified formulas
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This idea is formally called universal instantiation:
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These come in pairs for each quantifier (universal/existential)
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One is called generalization, the other one called instantiation
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CS311H: Discrete Mathematics
Inference Rules for Quantifiers
∀x .P (x )
(for any c)
P (c)
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Işıl Dillig,
Example
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Universal Generalization
Consider predicates man(x) and mortal(x) and the hypotheses:
1. All men are mortal:
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Suppose we can prove a claim for an arbitrary element in the
domain.
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Since we’ve made no assumptions about this element, proof
should apply to all elements in the domain.
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This correct reasoning is captured by universal generalization
2. Socrates is a man:
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Using rules of inference, prove mortal(Socrates)
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CS311H: Discrete Mathematics
Inference Rules for Quantifiers
P (c) for arbitrary c
∀x .P (x )
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Example
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Caveat About Universal Generalization
Prove ∀x .Q(x ) from the hypotheses:
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When using universal generalization, need to ensure that c is
truly arbitrary!
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If you prove something about a specific person Mary, you
cannot make generalizations about all people
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In a proof, this means c must be a fresh name not used
previously
1.
2.
3.
4.
5.
__________________________________
6.
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Inference Rules for Quantifiers
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2
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Existential Instantiation
Example Using Existential Instantiation
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Consider formula ∃x .P (x ).
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We know there is some element, say c, in the domain for
which P (c) is true.
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This is called existential instantiation:
Consider the hypotheses ∃x .P (x ) and ∀x .¬P (x ). Prove that we
can derive a contradiction (i.e., false) from these hypotheses.
1.
2.
3.
4.
5.
6.
∃x .P (x )
(for unused c)
P (c)
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Here, c is a fresh name (i.e., not used before in proof).
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Hypothesis
Hypothesis
Otherwise, can prove non-sensical things such as: ”There exists
some animal that can fly. Thus, rabbits can fly”!
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Işıl Dillig,
Existential Generalization
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Suppose we know P (c) is true for some constant c
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Then, there exists an element for which P is true
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Thus, we can conlude ∃x .P (x )
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This inference rule called existential generalization:
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Example Using Existential Generalization
Consider the hypotheses atUT (George) and smart(George).
Prove ∃x . (atUT (x ) ∧ smart(x ))
1.
2.
3.
4.
P (c)
∃x .P (x )
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Universal Instantiation
Universal Generalization
Existential Instantiation
Existential Generalization
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Example 1
Rule of Inference
∀x .P (x )
(anyc)
P (c)
P (c) (for arbitraryc)
∀x .P (x )
∃x .P (x )
P (c) for fresh c
P (c)
∃x .P (x )
Inference Rules for Quantifiers
Hypothesis
Hypothesis
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Name
atUT (George)
smart(George)
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Summary of Inference Rules for Quantifiers
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∃x .P (x )
∀x .¬P (x )
Prove that these hypotheses imply ∃x .(P (x ) ∧ ¬B (x )):
1. ∃x . (C (x ) ∧ ¬B (x )) (Hypothesis)
2. ∀x . (C (x ) → P (x )) (Hypothesis)
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Işıl Dillig,
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Example 2
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Example 3
Prove the below hypotheses are contradictory by deriving false
Is this formula valid, unsat, or contingent? Prove your answer!
1. ∀x .(P (x ) → (Q(x ) ∧ S (x ))) (Hypothesis)
((∀x .P (x )) ∧ (∀x .Q(x ))) → (∀x .(P (x ) ∧ Q(x )))
2. ∀x .(P (x ) ∧ R(x )) (Hypothesis)
3. ∃x .(¬R(x ) ∨ ¬S (x )) (Hypothesis)
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Example 4
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Example 4, cont.
What’s wrong with the following “proof” of validity?
Is the following formula valid, unsatisfiable, or contingent?
1. (∀x .(P (x ) ∨ Q(x ))) ∧ ¬(∀x .P (x ) ∨ ∀x .Q(x ))
(∀x .(P (x ) ∨ Q(x ))) → (∀x .P (x ) ∨ ∀x .Q(x ))
2. (∀x .(P (x ) ∨ Q(x )))
∧ -elim, 1
4. ∃x .¬P (x ) ∧ ∃x .¬Q(x )
5. ∃x .¬P (x )
7. ¬P (c)
∃-inst, 5
9. P (c) ∨ Q(c)
10. Q(c)
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Işıl Dillig,
∧-elim, 1
De Morgan, 3
6. ∃x .¬Q(x )
8. ¬Q(c)
∧-elim, 4
∃-inst, 6
∨-elim, 9, 7
11. Q(c) ∧ ¬Q(c)
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Some terminology
Formalizing statements in logic allows formal,
machine-checkable proofs
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But these kinds of proofs can be very long and tedious
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In practice, humans write slight less formal proofs, where
multiple steps are combined into one
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We’ll now move from formal proofs in logic to less formal
mathematical proofs!
CS311H: Discrete Mathematics
3. ¬(∀x .P (x ) ∨ ∀x .Q(x ))
∀-inst, 2
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Formal vs. Informal Proofs
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∧-elim, 4
premise
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4
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Theorem: A mathematical statement that has been proven
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Many famous mathematical theorems, e.g., Pythagorean
theorem, Fermat’s last theorem
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Pythagorean theorem: Let a, b the length of the two sides of
a right triangle, and let c be the hypotenuse. Then,
a 2 + b2 = c2
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Fermat’s Last Theorem: For any integer n greater than 2, the
equation a n + b n = c n has no solutions for non-zero a, b, c.
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Theorems, Lemmas, and Propositions
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There are many correct mathematical statements, but not all
of them called theorems
Less important statements that can be proven to be correct
are propositions
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Another variation is a lemma: minor auxiliary result which
aids in the proof of a theorem/proposition
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Corollary is a result whose proof follows immediately from a
theorem or proposition
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Conjectures vs. Theorems
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Conjecture is a statement that is suspected to be true by
experts but not yet proven
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Goldbach’s conjecture: Every even integer greater than 2 can
be expressed as the sum of two prime numbers.
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One of the oldest unsolved problems in number theory!
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General Strategies for Proving Theorems
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To prove p → q in a direct proof, first assume p is true.
Direct proof: p → q proved by directly showing that if p is
true, then q must follow
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Then use rules of inference, axioms, previously shown
theorems/lemmas to show that q is also true
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Proof by contraposition: Prove p → q by proving ¬q → ¬p
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Example: If n is an odd integer, than n 2 is also odd.
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Proof by contradiction: Prove that the negation of the
theorem yields a contradiction
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Proof by cases: Exhaustively enumerate different possibilities,
and prove the theorem for each case
Proof: Assume n is odd. By definition of oddness, there must
exist some integer k such that n = 2k + 1. Then,
n 2 = 4k 2 + 4k + 1 = 2(2k 2 + 2k ) + 1, which is odd. Thus, if
n is odd, n 2 is also odd.
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Observe: This proof implicitly uses universal generalization
and existential instantiation (where?)
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In many proofs, one needs to combine several different strategies!
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More Direct Proof Examples
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Another Example
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An integer a is called a perfect square if there exists an
integer b such that a = b 2 .
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Example: Prove that if m and n are perfect squares, then mn
is also a perfect square.
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Direct Proof
Many different strategies for proving theorems:
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Example: Prove that every odd number is the difference of
two perfect squares.
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