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CM10196
Lecture Notes 2
Issued 12 October 2016
1.
Commutative laws
Let x, y be propositions. Then x ∧ y is true if
Commutative law for conjunction.
and only if y ∧ x.
Equivalent formulation. For variables X, Y the Boolean formula
(X ∧ Y ) ≡ (Y ∧ X)
is identically true.
Proof.
Straightforward check of tables defining ∧ and ≡.
Commutative law for disjunction.
only if y ∨ x.
Let x, y be propositions. Then x ∨ y is true if and
Equivalent formulation. For variables X, Y the Boolean formula
(X ∨ Y ) ≡ (Y ∨ X)
is identically true.
Proof.
2.
Straightforward check of tables defining ∨ and ≡.
Associative laws
In what follows we will formulate each law only in a form of identically true Boolean
formula.
Associative law for conjunction. For variables X, Y , Z the Boolean formula
((X ∧ Y ) ∧ Z) ≡ (X ∧ (Y ∧ Z))
is identically true.
Associative law for disjunction.
For variables X, Y , Z the Boolean formula
((X ∨ Y ) ∨ Z) ≡ (X ∨ (Y ∨ Z))
is identically true.
Proof.
3.
Straightforward check of tables.
Distributive laws
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Conjunction over disjunction.
For variables X, Y , Z the Boolean formula
(X ∧ (Y ∨ Z)) ≡ ((X ∧ Y ) ∨ (X ∧ Z))
is identically true.
Disjunction over conjunction.
For variables X, Y , Z the Boolean formula
(X ∨ (Y ∧ Z)) ≡ ((X ∨ Y ) ∧ (X ∨ Z))
is identically true.
Proof.
4.
Straightforward check of tables.
De Morgan’s laws
For variables X, Y , Z the Boolean formulae
¬(X ∧ Y ) ≡ (¬X ∨ ¬Y ),
¬(X ∨ Y ) ≡ (¬X ∧ ¬Y ).
are identically true.
Proof.
Straightforward check of tables.
5.
Some other laws of logic
(i)
(ii)
(iii)
(iv)
(v)
X ∨ ¬X;
¬¬X ≡ X;
¬(X ∧ ¬X);
((¬X → Y ) ∧ (¬X → ¬Y )) → X;
(X → Y ) ≡ (¬X ∨ Y ).
Observe that De Morgan’s laws combined with (ii) allow to express ∧ via ∨ and vice
versa. For example, (ii) implies that a disjunction U ∨ V , where U , V are variables, can be
replaced by an equivalent formula ¬¬U ∨ ¬¬V , which in turn, by the first of De Morgan’s
laws is equivalent to ¬(¬U ∧ ¬V ).
Also, (v) provides a way of replacing → by ∨.
It follows that every Boolean formula is equivalent to a formula having only one of
the following pairs of connectives:
(a) ¬, ∧ or
(b) ¬, ∨ or
(c) ¬, →.
6.
Reductio ad absurdum. Direct and inverse theorems
The tautology
((¬X → Y ) ∧ (¬X → ¬Y )) → X
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(formula (iv) from Section 5) is often used for proving mathematical statements by “leading
to contradiction”. To prove a proposition x it’s sufficient to assume the negation ¬x of x
and deduce from ¬x two contradictory statements, say, y and ¬y. The the above tautology
implies that x is true.
Many mathematical theorems can be formally described by an implication x → y.
Call this implication direct theorem. Then the proposition y → x is known as inverse
theorem. Even if the direct theorem is true, the inverse may or may not be true. The
proposition ¬x → ¬y is called opposite theorem. The proposition ¬y → ¬x is, therefore,
opposite to inverse theorem. Observe, that an opposite to inverse theorem is true if and
only if the direct theorem is true, because of the tautology
(X → Y ) ≡ (¬Y → ¬X).
In some cases it’s easier to prove opposite to inverse theorem than the direct theorem.
If a theorem has a structure of an implication x → y, then x is called the sufficient
condition of the theorem, while y is its necessary condition. If a theorem has a form “x
is necessary and sufficient for y” it means that two statements are true at the same time:
x → y and y → x.
7.
Disjunctive normal form of a Boolean formula
Let X1 , X2 , . . . , Xn be variables. Yi is called a literal if Yi is either Xi or ¬Xi for any
i.
A conjunctive term is a conjunction of the kind Yi1 ∧ Yi2 ∧ · · · ∧ Yim , where ij is one
of the numbers 1, 2, . . . , n for 1 ≤ j ≤ m, and each Yij is a literal.
Definition
A Boolean formula F is in disjunctive normal form (DNF) if F is of the form
F1 ∨ F2 ∨ · · · ∨ Fk ,
where Fi is a conjunctive term for any i.
Theorem For each Boolean formula G with variables X1 , X2 , . . . , Xn there exists a
Boolean formula F in DNF, with variables from {X1 , X2 , . . . , Xn }, such that G ≡ F .
We don’t prove this theorem in this course.
Examples
(1) Let G be the formula X → Y . From (v), Section 4 it follows that G ≡ ¬X ∨ Y .
The righthand side of the latter equivalence is a formula in disjunctive normal form with
conjunctive terms ¬X and Y .
(2)
Let G be the formula
((¬X → Y ) ∧ (¬X → ¬Y )) → X.
Using again (v), Section 4 and De Morgan’s laws we can write a following string of equivalent formulae:
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((¬X → Y ) ∧ (¬X → ¬Y )) → X
¬((X ∨ Y ) ∧ (X ∨ ¬Y )) ∨ X
¬(X ∨ Y ) ∨ ¬(X ∨ ¬Y ) ∨ X
(¬X ∧ ¬Y ) ∨ (¬X ∧ Y ) ∨ X.
The last formula in this list is equivalent to G and is in DNF.
8.
Conjunctive normal form of a Boolean formula
Conjunctive normal form (CNF) of a Boolean formula is a representation dual to its
representation in DNF.
A disjunctive term is a disjunction of the kind Yi1 ∨ Yi2 ∨ · · · ∨ Yim where ij is one of
the numbers 1, 2, . . . , n for 1 ≤ j ≤ m.
Definition
A Boolean formula F is in conjunctive normal form (CNF) if F is of the form
F1 ∧ F2 ∧ · · · ∧ Fk ,
where Fi is a disjunctive term for any i.
Theorem For each Boolean formula G with variables X1 , X2 , . . . , Xn there exists a
Boolean formula F in CNF, with variables from {X1 , X2 , . . . , Xn }, such that G ≡ F .
We don’t prove this theorem.
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