Chapter 2. Fundamentals of Logic.

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Chapter 2. Fundamentals of Logic.

• The rules of logic give precise meaning to
  mathematical
• statements.
• We need to study logic to learn how to construct
  valid
• reasoning.
• Logic focuses on the relationship between
  propositions
• (not the content of any particular
  proposition)
• Logical methods are used in mathematics to prove
  theorems
• and in computer science to prove that programs
  do what they
• supposed to do

Definition. A proposition (or a statement) is a
sentence that is either true or false,
but not both. (boolean variables)
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• Which of the following are propositions?
• Two is a prime number.
• How are you?
• x12.
• There is life on Saturn
• n2n41 is a prime number for all natural
  numbers,
• ?n Î N
• (like any statement, this one relies upon
  some definitions,
• terminology and notations)

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Logic is the algebra of propositions or Boolean
algebra

• Let p, q, r, be propositions.
• Then we can define binary operators (connectives
  )
• ? (and)
• ? (inclusive or)
• ? (implication)
• ? (biconditional)
• and unary operator
• ? (negation)

A binary operator takes two propositions to form
a compound proposition. The truth value of this
compound proposition is defined by input
primitive propositions and by the operator.
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Conjunction s pÙq (p and q) is true when
both p and q are true and false otherwise. p
grass is green. q horses like oats. s (p Ù
q) grass is green and horses like oats When p, q
are two true statements, (p Ù q) is true. If p
is true and q is false, (p Ù q) is false If both
p and q are false, (p Ù q) is false
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Disjunction p?q (p or q) is true when
either p or q or both are true ( inclusive or )
and false otherwise.

• p q s p ? q
• 1 1 1
• 1 0 1
• 0 1 1
• 0 0 0

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Implication p ? q Terminology p
antecedent, hypothesis, premises.
q consequent, conclusion.

• p implies q
• if p then q
• p is sufficient (condition) of q
• q whenever p
• p only if q
• q is necessary (condition) of p

Note p ? q does NOT imply ? p ? ? q, i. e. p is
sufficient, but not necessary condition of q.
If the weather is good I will go for a walk.
The weather is good ? I will go for a walk does
not imply that I go for a walk ? the weather is
good or that the weather is bad ? I will not go
for a walk
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• Name the antecedent p and consequent q, p ?q, in
  each
• of the following statements.
• If Peter gets scholarship he will go to college.
• A sufficient condition for using 6 storage
  locations is that
• a 2?3 array is to be stored.
• Susan will pass her physics class only if she
  studies hard.
• Good combustion is a necessary condition for
• high gasoline mileage.

q
p
q
p
p
q
q
p
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Implication p ? q has a truth value It is
defined to be true in all cases except when p is
true and q is false.
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p ? q is NOT equivalent to q ? p (converse of p
? q ) since not for all assignments of p, q truth
value of p ? q is the same as truth value of q
? p
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Definition A compound statement s(p, q, r,)
that is always true, no matter what the truth
values of p, q, r,. is called a tautology. A
compound statement s(p, q, r,) that is always
false is called a contradiction Definition Two
compound statements s1(p, q, r,) and s2(p, q,
r,) are called equivalent, if s1? s2, is a
tautology. The notations s1 ? s2 denotes that s1
and s2 are logically equivalent.
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Biconditional p ? q (or p ? q) is true when p
and q have the same truth value and false
otherwise.

• p is sufficient and necessary condition of q
• q if and only if (iff) p
• p and q are equivalent

• p q p ? q q ? p (p ? q)?(q ? p)
  p ? q r
• 1 1 1 1 1 1 1
• 1 0 0 1 0 0 1
• 0 1 1 0 0 0 1
• 0 0 1 1 1 1 1

(p ? q) ?( q ? p) ? p ?q, or r (p ? q) ?( q ?
p) ? p ?q is a tautology.
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Negation ? p (not p )
p ? p 1 0 0 1