2013
Logic &
proofs
BY:Nauf Almarwani
Chapter 1
Introduction to Discrete Mathematics
#Discrete Mathematics: is the part of mathematics devoted to the
study of discrete objects.
.‫ هو الجزء من الرياضيات المكرسة لدراسة الكائنات منفصلة‬:‫الرياضيات المتقطعة‬
#Mathematical Logic: is a tool for working with elaborate compound
statements.
.‫المنطق الرياضي هو أداة للعمل مع وضع البيانات المركبه‬
#Propositional Logic: is the logic of compound statements built from
simpler statements using so-called Boolean connectives.
.‫ هو منطق من البيانات المركبه بنيت من بيانات أبسط باستخدام ما يسمى أدوات الوصل المنطقية‬:‫منطق اقتراحي‬
1.1 Propositional
Logic:
A proposition: is a declarative sentence (that is, a sentence that
declares a fact) that is either true
or false, but not both.
.‫ ولكن ليس االثنين معا‬،‫ هو الجملة البيانيه (الجملة التي تعلن حقيقة) التي هي إما صحيحة أو خاطئة‬:‫االقتراح‬
EXAMPLE 1:
All the following declarative sentences are propositions.
1 . Washington, D.C., is the capital of the United States of America. (T)
2 . Toronto is the capital of Canada. (F)
3 . 1 + 1 = 2 . (T)
4. 2 + 2 = 3 . (F)
Command and questions are not propositions
.‫االوامر واألسئلة ليست مقترحات‬
EXAMPLE 2:
Consider the following sentences.
1 . What time is it?
2 . Read this carefully.
3.x+1=2.
4 . x + y = Z.
‫في المثال االول اعطانا جمل بيانيه فحددنا هي صائبه ام‬
‫خاطئه‬
.‫ وهي صح‬2=1+1 -3 ‫مثال رقم‬
‫في المثال الثاني اعطانا اسئله و اوامر وليس جمل بيانيه‬
. ‫فما نقدر نحدد هيا صح او غلط‬
we use letters to denote propositional variables (or statement
variables)
.‫نستخدم الحروف للداللة على متغيرات اقتراحيه او متغيرات بيانيه‬
A proposition: (denoted p, q, r, …) is simply:a statement (i.s., a
declarative sentence)
.‫) وهي جمله بيانيه‬p,q.,r( ‫نرمز لالقتراح بالرمز‬
The area of logic that deals with propositions is called the
propositional calculus or propositional logic.
‫وتسمى منطقة المنطق الذي تتعامل مع المقترحات حساب التفاضل والتكامل االقتراحيه أو المنطق االقتراحي‬
*Some Popular Boolean Operators.
Formal Name
Negation operator
Conjunction operator
Disjunction operator
Exclusive-OR operator
Implication operator
Biconditional operator
Nickname
NOT
AND
OR
XOR
IMPLIES
IFF
Arity
Unary
Binary
Binary
Binary
Binary
Binary
Symbol
¬




↔
DEFINITION 1
Let p be a proposition.
The negation of p, denoted by⌐p
(also denoted by −p), is the
statement.
"It is not the case that p."
The proposition ⌐p is read "not
p"
The truth value of the negation
of p, ⌐p, is the opposite of the
truth value of p.
.‫ اقتراح‬p ‫لنجعل‬
–p ‫⌐ او نرمز له بالرمز‬p ‫ هو‬p ‫نفي‬
p ‫ويقرا ليس‬
p ‫ هي عكس القيمه الحقيقيه ل‬p ‫القيمه الحقيقيه للنفي‬
EXAMPLE 3
Find the negation of the proposition
"Today is Friday."
Solution:
"Today is not Friday,"
or
"It is not Friday today."
‫المثال الثالث‬
‫ اليوم الجمعه‬:‫طلب نفي الجمله‬
"‫ببساطه الحل "اليوم ليس الجمعه‬
The truth table
p
T
F
p
F
T
DEFINITION 2
Let p and q be propositions.
The conjunction of p and q,
denoted by pq, is the
proposition "p and q"
The conjunction p  q is true
when both p and q are true and
is false otherwise..
q ‫ و‬p ‫لنجعل‬
‫مقترحين‬
‫لجمع المقترحين معا‬
 ‫نرمز له بالرمز‬
‫اذا كان جميعهما‬
‫صائب فيكون الناتج‬
‫صائب‬
‫لو كان واحد منهم‬
‫خاطئ جميعها خاطئ‬
The truth table
p
F
F
T
T
EXAMPLE 5:
Find the conjunction of the propositions p and q
where p is the proposition "Today is Friday"
and q is the proposition "It is raining today."
p
T
T
F
F
q
T
F
T
F
pq
T
F
F
F
p
F
F
T
T
q
F
T
F
T
q
F
T
F
T
pq
F
F
F
T
pq
F
F
F
T
The truth table
DEFINITION 3
Let p and q be
propositions.
The disjunction of p and
q, denoted by p V q, is
the proposition
"p or q ." The
disjunction p V q is
false when both p and q
are false and is true
otherwise.
‫ مقترحين‬q ‫ و‬p ‫لنجعل‬
‫لنفصل المقترحين‬
‫لو كان واحد من القيم صائب‬
‫كلها صائب‬
‫لو كانت كلها خاطئه اذن‬
‫خاطئ‬
EXAMPLE 6:
Find the Disjunction of the propositions p and q
where p is the proposition "Today is Friday"
and q is the proposition "It is raining today."
p
T
q
T
pq
T
p
F
F
T
T
q
F
T
F
T
pq
F
T
T
T
T
F
F
F
T
F
T
T
F
The truth table
DEFINITION 4
Let p and q be
propositions.
The exclusive or of p and
q, denoted by pq, is the
proposition that is true
when exactly one of p and q
is true and is false
otherwise.
‫ مقترحين‬q ‫ و‬p ‫لنجعل‬
‫االستثناء للمقترحين اذا كان‬
‫فقط واحد منهم صائب‬
‫لو كان االثنين خاطئ او‬
‫صائب اذا خاطئ‬
Conditional Statements:
p
F
F
T
T
q pq
F F
T T
F T
T F
The truth table
DEFINITION 5
Let p and q be propositions.
‫ مقترحين‬q ‫ و‬p ‫لنجعل‬
The conditional statement p 
‫العباره الشرطيه لهم‬
q is the proposition "if p,
q ‫ اذا‬p ‫اذا‬
then q ."
‫العباره الشرطيه تكون خاطئه فقط اذا‬
The conditional statement p
‫ خاطئ والبقية صائب‬p ‫كان‬
q is false when p is true and
‫ تسمى فرضيه‬p ‫في العباره الشرطيه‬
q is false, and true otherwise.
.‫ تسمى النتيجه‬q‫و‬
In the conditional statement p
 q , p is called the
hypothesis (or antecedent or
premise) and q is called the
conclusion (or consequence).
p
F
F
T
T
q pq
F
T
T T
F
F
T T
EXAMPLE 7:
Let p be the statement "Maria learns discrete mathematics" and q the statement
"Maria will find a good job." Express the statement p  q as a statement in English.
p
q
pq
T
T
T
7 ‫مثال‬
T
F
T
F
T
F
‫يطلب العباره الشرطيه‬
F
F
T
‫الجمل هي اذا "ماري تعلمت الرياضيات المنفصله" اذا‬
"‫"بتالقي وظيفه جيده‬
‫كل االحتماالت صحيحه لكن لو ماتعلمت الرياضيات م‬
"‫بتالقي اي وظيفه جيده‬
EXAMPLE 8
What is the value of the variable x after the statement
if 2 + 2 = 4 then x := x + 1
Solution:
Because 2 + 2 = 4 is true, the assignment statement x := x + 1 is
executed. Hence, x has the value 0 + 1 = 1 after this statement is
encountered.
8 ‫في المثال‬
#English Phrases Meaning p  q
‫اعطانا قيمة االول صحيحه والثانيه نعوض طبعا‬
• “p implies q”
‫م بنقول القيمه خاطئه الن االولى صائبه اذا‬
• “if p, then q”
‫الجميع صائب‬
• “if p, q”
• “when p, q”
• “whenever p, q”
• “q if p”
• “q when p”
• “q whenever p”
• “p only if q”
• “p is sufficient for q”
• “q is necessary for p”
• “q follows from p”
• “q is implied by p”
We will see some equivalent logic expressions later.
Converse, Inverse, Contrapositive
Its converse
Its inverse
Its contrapositive
q  p
¬p  ¬q
¬q  ¬ p
EXAMPLE 9
What are the contrapositive, the converse, and the inverse of the conditional statement
"The home team wins whenever it is raining."?
Whenever = pq
So
"If it is raining, then the home team wins."
are the contrapositive
the converse
the inverse
"If the home team
"If the home team
"If it is not raining,
does not win, then
wins, then it is
then the home team
it is not raining."
raining."
does not win."
The truth table
DEFINITION 6
Let p and q be propositions.
The biconditional statement p ↔
‫ مقترحين عبارتين‬q ‫ و‬p ‫لنجعل‬
q is the proposition "p if and
‫شرطيتين‬
only if q ." The biconditional
‫اذا واذا فقط تكون صائبه‬
statement p ↔ q is true when p
‫اذا جميعهما صائب او جميعهما‬
and q have the same truth
.‫خاطئ‬
values, and is false otherwise.
Biconditional statements are
also called bi-implications.
p
F
F
T
T
q
F
T
F
T
EXAMPLE 10
Let p be the statement "You can take the flight" and let q be the
statement "You buy a ticket."
p  q
T
F
F
T
Then p ↔ q is the statement
P
q
"You can take
"You buy a
the flight"
ticket."
T
T
T
F
F
T
F
F
p ↔ q
T
F
F
T
Truth Tables of Compound Propositions:
‫البيانات المركبه من الجداول مقترحه‬
Construct the truth table of the compound proposition
(p  ⌐q) (p  q).
p
Q
⌐q
p  ⌐q
p  q
(p  ⌐q) (p  q)
T
T
F
T
T
T
T
F
T
T
F
F
F
T
F
F
F
T
F
F
T
T
F
F
Precedence of Logical Operators :
‫أسبقية العوامل المنطقية‬
Operator
Precedence
⌐



↔
1
2
3
4
5
Translating English Sentences:
EXAMPLE 12
How can this English sentence be translated into a logical expression?
"You can access the Internet from campus only if you are a computer science major or
you are not a freshman."
‫نقسم الكالم اول شي‬
You can access the Internet from
P (q⌐r)
campus= P
‫يعني يمكنك الوصول الى االنترنت من الجامعه‬
only if you= 
you are a computer science major= q
‫اذا كنت عالم كمبيوتر‬
Or =
are not a freshman= ⌐r
‫اذا كنت غير مبتدئ‬
EXAMPLE 13
How can this English sentence be translated into a logical expression?
"You cannot ride the roller coaster if you are under 4 feet tall unless you are older than
16 years old."
p  ⌐q⌐r
System Specifications:
EXAMPLE 14
Express the specification "The automated reply cannot be sent when the file system is
full" using logical connectives.
p⌐q
Boolean Searches:
p
q
pq
pq
pq
pq
p↔q
⌐p
T
T
T
T
F
T
T
F
T
F
F
T
T
F
F
F
F
T
F
T
T
T
F
T
F
F
F
F
F
T
T
T
Logic and Bit Operations
Truth Value
Bit
1
T
0
F
A bit string is a sequence of zero or more
bits. The length of this string is the number
of bits in the string.
‫ طول هذه السلسلة هو عدد‬.‫سلسلة بت هو سلسلة من صفر أو أكثر من البتات‬
.‫البتات في السلسلة‬
EXAMPLE 21
Find the bitwise OR, bitwise AND, and bitwise XOR of the bit strings 0 1 1 0 1 1 0 1 1 0 and
1 1 000 1 1 1 0 1 . (Here, and throughout this book, bit strings will be split into blocks of
four bits to make them easier to read.)
0110110110
1100011101
OR
1110111111
AND
0100010100
XOR
1010101011
1.2 Propositional Equivalence:
‫اقتراح التكافؤ‬
A compound proposition that is always true, no matter what the truth values of the
propositions that occur in it, is called a tautology. A compound proposition that is
always false is called a contradiction. A compound proposition that is neither a
tautology nor a contradiction is called a contingency.
‫التقرير المركب هو دائما صحيح بغض النظر عن القيم الحقيقة من المقترحات التي تحدث فيه يسمى تكرار الكلمه ويطلق‬
. ‫ ويطلق على اقتراح المركب الذي ليس تكرار وال تناقض احتمال‬.‫على اقتراح المركب الذي هو دائما كاذب تناقض‬
1- tautology:
p ⌐p ‫كل قيمه صائبه وقانونه‬
P
T
F
⌐p
F
T
p ⌐p
T
T
2- contradiction:
P⌐P ‫كل قيمه خاطئه وقانونه‬
P
T
F
3- Contingency:
pq
# De Morgan's Laws
⌐ (pq)  ⌐p  ⌐q
⌐(pq)  ⌐p⌐q
⌐p
F
T
p ⌐p
F
F
#Logical Equivalences:
The compound propositions p and q are called logically equivalent if p ↔ q is a
tautology.
The notation p  q denotes that p and q are logically equivalent.
‫التقرير المنطقي يسمى التكافئ المنطقي اذا واذا فقط كان صائب‬
The symbol  is not a logical connective and p  q is not a compound
proposition
‫ ليس تقرير مركب‬p  q ‫ ليس رابط منطقي‬ ‫الرمز‬
but rather is the statement that p ↔ q is a tautology
‫ صائب‬p ↔ q‫التقرير‬
The symbol ↔ is sometimes used instead of  to denote logical equivalence.
..‫ للدالله على التكافؤ المنطقي‬ ‫الرمز↔ احيانا يستخدم بدل‬
EXAMPLE 2
Show that ⌐(pq ) and ⌐p⌐q are logically equivalent.
‫في السؤال اعطانا رمزين وبيقول هل هما متكافئين منطقيا؟‬
‫نرسم جدول‬
p
q
pq
⌐(pq )
⌐p
⌐q
⌐p⌐q
T
T
F
F
T
F
T
F
T
T
T
F
F
F
F
T
F
F
T
T
F
T
F
T
F
F
F
T
EXAMPLE 3:
Show that p  q and ⌐p  q are logically equivalent ?
p
T
T
F
F
Q
T
F
T
F
pq
T
F
T
T
⌐p
F
F
T
T
⌐pq
T
F
T
T
EXAMPLE 4:
Show that p(qr) and (p q)  (p  r) are logically equivalent. This is the
distributive law of disjunction over conjunction?
8=23 =‫هنا اعطانا ثالث قيم‬
p
q
r
qr p(qr) (p q) (p  r) (p q)  (p  r)
T T T
T
T
T
T
T
T T F
F
T
T
T
T
T F T
F
T
T
T
T
T
F
F
F
F
F
T
T
F
F
F
T
F
T
F
F
T
F
F
F
T
T
F
F
F
T
T
T
F
F
T
T
F
T
F
T
T
F
F
F
Logical Equivalences:
‫المكافئات المنطقية‬
name
Identity
laws
Domination Idempotent Double
laws
negation
laws
law
Commutative
laws
equivalence
p  T  p
p  F  p
p  p  p
p  q  q p
p  F  p
p  T  p
p  p  p
name
Associative Distributive
laws
laws
equivalence
(p  q)  r 
p(q  r)
(p  q)  r 
p (q r)
p  (q  r)  (p
 q)  (p  r)
p  (q  r) (p 
q)  (p  r)
De
Morgan's
laws
⌐(p  q)  ⌐p
 ⌐q
⌐(p  q)  ⌐p
 ⌐q
⌐(⌐p)  p
p  q  q p
Absorption Negation
laws
laws
P  (p  q)  p
p (p  q)  p
p  ⌐p  T
p ⌐p  F
Logical Equivalences Involving Conditional Statements
‫المكافئات المنطقية التي تنطوي على البيانات الشرطيه‬
p q  ⌐p  q
p  q  ⌐q ⌐p
p  q  ⌐p  q
p  q  ⌐(p  ⌐ q)
⌐ (p  q) P  ⌐q
(p  q)  (p  r)  p  (q  r)
(p  r ) (q  r)  (p  q)  r
(p  q)  (p  r)  p (q  r )
(p  r)  (q  r)  (p q)  r
Logical Equivalences Involving Biconditionals:
P ↔ q  (p  q) (q  p)
P ↔ q  ⌐p ↔ ⌐q
P ↔ q  (p q)  (⌐p ⌐q)
⌐ (p ↔ q)  P ↔ ⌐q
Using De Morgan's Laws:
EXAMPLE 5
Use De Morgan's laws to express the negations of "Migue1 has a cellphone and he has a laptop
computer" and "Heather will go to the concert or Steve will go to the concert."
p= "Migue1 has a cellphone"
q=" has a laptop computer"
‫اول شيء طالب النفي‬
(pq)=⌐ (pq) = ⌐p  ⌐q (De Morgan's laws)
r ="Heather will go to the concert"
s= "Steve will go to the concert."
OR ‫بينهم‬
(rs)=⌐ (rs)= ⌐r  ⌐s
(De Morgan's laws)
#Constructing New Logical Equivalences:
‫انشاء متكافئات منطقية جديده‬
‫هنا ممكن ننشئ قوانين من قوانين‬
EXAMPLE 6
Show that ⌐ ( p  q) and p ⌐q are logically equivalent.
‫طالب مننا دليل على انها متساويه منطقيا‬
‫نأخذ الطرف االول‬
⌐ ( p  q)  ⌐(⌐pq) ‫اثبتنا انها متساويه في التمرين الثالث‬
 ⌐(⌐p)⌐q (De Morgan's laws)
 p ⌐q Double negation law
EXAMPLE 7
Show that ⌐(p  ( ⌐ p  q)) and ⌐p  ⌐q are logically equivalent by developing a
series of logical equivalences.
⌐(p  ( ⌐ p  q))  ⌐p⌐(⌐pq) by the second De Morgan law
 ⌐p  [ ⌐ ( ⌐ p )  ⌐ q ] by the first De Morgan law
 ⌐p(p⌐q) by the double negation law
 (⌐pp)  (⌐p ⌐q) by the second distributive law
 F (⌐p ⌐q) because (⌐pp)  F
 (⌐p ⌐q)  F by the commutative law for disjunction
 (⌐p ⌐q)
by the identity law for F
⌐(p  ( ⌐ p  q))  ⌐p  ⌐q
EXAMPLE 8
Show that (p q)  (p  q) is a tautology?
(p q)  (p  q)  ⌐(p q)  (p  q)
 (⌐p  ⌐q)  (p  q) by the first De Morgan law
 (⌐p  p) (⌐q q)
by the associative and commutative laws for
disjunction
 T  T by Example I and the commutative
law for disjunction
T
by the domination law
1.3 Predicates and Quantifiers
‫الخبر والكميه‬
1- Predicates:
Statements involving variables, such as
"x > 3 , " "x = y + 3," "x + y = z,"
"computer x is under attack by an intruder,"
and
"computer x is functioning properly,"
These statements are neither true nor false when the
values of the variables are not specified.
EXAMPLE 1:
Let P(x) denote the statement "x > 3."
What are the truth values of P (4) and
P (2)?
‫ صحيحه في المعادله‬X ‫المطلوب هنا متى تكون قيمة‬
X>3
4 ,2 ‫ وهما‬X ‫طبعا اعطانا قيمتين ل‬
: ‫نعوض‬
4>3…..TRUE
2>3…..FALSE
: ‫ بيانات تشمل متغيرات مثل‬: ‫الخبر‬
X>3
)variable( ‫ متغير‬X
)Predicates( ‫> الخبر‬3
: ‫مثال‬
.‫ تعرض لفايروس من قبل متسلل‬x ‫الكمبيوتر‬
. ‫ يعمل بشكل صحيح‬x ‫و الكمبيوتر‬
‫هذه البيانات ليست صحيحه وال خاطئه لماذا؟‬
‫بسبب انه اليوجد له قيمه محدده‬
‫) مجهول‬x( ‫يعني‬
: ‫عرفنا ان‬
X is a variable
> 3 is a Predicates
‫نقول أن‬
"x is greater than 3" by
P (x)
the value of the propositional function P at x
‫ هو الخبر‬3 ‫ و اكبر من‬X ‫باختصار عندنا متغير‬
3‫ اكبر من ال‬X ‫لمن نقول ان ال‬
X ‫ في‬P ‫ تعني قيمه الداله المقترحه‬P (X) ‫نختصرها ب‬
P = Predicates
EXAMPLE 2:
Let A (x) denote the statement "Computer x is under attack by an intruder."
Suppose that of the computers on campus, only CS2 and MATH 1 are currently
under attack by intruders. What are truth values of A (CS l), A(CS2), and
A(MATH l )?
: ‫شرح السؤال‬
‫ تعرض للهجوم من قبل متسلل‬X ‫= الكمبيوتر‬A (x)
‫( فقط‬MATH 1) ‫( و‬CS2) ‫االجهزه التي تعرضت للهجوم في الجامعه هي‬
‫ ؟؟‬A(CS l), A(CS2), A(MATH l ) ‫ ما هي القيم الحقيقة‬: ‫المطلوب‬
: ‫الحل‬
A (CS2) and A (MATH l ) are true
... ‫طبعا هما جهازين تعرضو للهجوم‬
A (CS I ) is false
.... ‫الجهاز الثالث ليس مذكور انه تعرض للجوم‬
EXAMPLE 3:
Let Q (x , y) denote the statement "x = y + 3 ." What are the truth values of the
propositions
Q(1 , 2) and Q(3, O)?
x = y + 3 : ‫اول شي عندنا المعادله‬
X , Y ‫نعوض بقيمة‬
Q(1 , 2) ‫مرا بـ‬
Q(3, O) ‫ومرا بـ‬
x = y +3
1 = 2+3 >>> FALSE ..
x = y +3
3 = 0 + 3>>> TRUE
A statement of the form P (XI, X2, • • . , Xn) is the value of the propositional function P
at the n-tuple (Xl, X2, . • • , xn), and P is also called a n -place predicate or a n-ary
predicate.
the n-tuple (Xl, X2, . • • , xn), ‫ في‬P ‫ هو قيمة الدالة المقترحه‬P (XI, X2, • • . , Xn) ‫البيانات التي بصورة‬
a n-ary predicate ‫ او‬n -place predicate ‫ويسمى أيضا‬
2- Quantifiers
When the variables in a propositional function are assigned values, the resulting
statement becomes a proposition with a certain truth value.
. ‫عندما تكون المتغيرات بقيم حقيقيه تكون النتيجه مقترحه مع بعض القيم الصحيحه‬
quantification, to create a proposition from a propositional function.
Quantification expresses the extent to which a predicate is true over a range of
elements. In English, the words all, some, many, none, and few are used in
quantifications
.‫ انشاء مقترح من داله مقترحه‬: ‫التقدير الكمي‬
.. ‫التقدير الكمي يعبر عن مدى صحة الخبر من مجموعة العناصر‬
..‫ وقليلة تستخدم لتقدير الكميه‬،‫ ال شيء‬،‫ كثير‬،‫ بعض‬،‫ كل‬.. ‫ الكلمات‬،‫في اللغة اإلنجليزية‬
two types of quantification
‫نوعين من التقدير الكمي‬
universal quantification
existential quantification
‫تقدير عالمي‬
‫تقدير وجودي‬
which tells us that a predicate is which tells us that there is one or
true for every element under
more element under consideration
consideration
for which the predicate is true.
‫يعني انه يوجد واحد او اكثر من العناصر صحيح يعني ان جميع خبر العناصر هو صحيح‬
The area of logic that deals with predicates and quantifiers is called the predicate
calculus
‫منطقة المنطق التي تتعامل مع الخبر والتقدير الكمي تسمى حساب التفاضل والتكامل‬
---------------------------------------------------------------------------------------------------------------: ‫معلومه مهمه‬
X ‫ محدد قيم‬DOMAIN ‫ الزم نعرف انه البد من وجود مجال‬universal quantification‫في ال‬
1- The universal quantification of P (x) is the statement
"P(x) for all values of x in the domain."
.‫ في المجال‬x ‫ لجميع قيم‬P(x)
The notation x P (x) denotes the universal quantification of P(x).
‫ داللة على التقدير الكمي العالمي‬x P (x) ‫الرمز‬
 is called the universal quantifier..
. ‫يسمى محدد الكمية العالمي‬
We read x P(x) as "for all x P(x)" or "for every x P(x )."
‫لكل‬... ‫نقرأ‬
An element for which P(x) is false is called a counterexample of
x P (x).
‫ خاطئ نسميه ب العنصر المضاد لكل عناصر المجال‬P(x) ‫عندما يكون عنصر‬
EXAMPLE 8:
Let P (x) be the statement "x + 1 > x." What i s the truth value of the quantification  x
P(x), where the domain consists of all real numbers?
Because P (x) is true for all real numbers x, the quantification
x P (x) is true.
‫ بجميع االرقام الحقيقه ستكون القيمه صحيحه‬x + 1 > x ‫يعني لو عوضنا‬
Quantifiers
9 ‫مثال لو اخذنا الرقم‬
9 ‫ وهي اكبر من‬10=1+9 ‫ اذا‬9 + 1 > 9
‫اذا صحيحه‬
Statement
 x P(x)
When True?
P(x) is true for every x.
ᴟx P(x)
There is an x for which P(x)
is true.
When False?
There is an x for which P(x)
is false.
P (x) is false for every x.
EXAMPLE 9
Let Q(x) be the statement "x < 2." What is the truth value of the quantification x Q
(x), where the domain consists of all real numbers?
‫لو عوضنا هنا بجميع االرقام الحقيقه نبدأ مثال من الصفر‬
x<2
0<2….true
1<2…true
2<2…false
3<2….false
x Q(x) ‫اذا‬
is false.
DEFINITION 2
The existential quantification of P(x) is the proposition "There exists an element x in
the domain such that P (x )."
P(x) ‫ هو مقترح "يوجد عنصر في المجال‬P(x) ‫العنصر الوجودي ل‬
We use the notation ᴲx P(x) for the existential quantification of P(x). Here ᴲ is called
the existential quantifier.
.‫ ويسمى محدد الكمية الوجودي‬P(x)‫ للتقدير الوجودي لل‬ᴲ ‫نستخدم الرمز‬
EXAMPLE 14
Let P (x) denote the statement "x > 3 ." What is the truth value of the quantification ᴲx P
(x), where the domain consists of all real numbers?
EXAMPLE 15:
‫هنا طال مننا نطلع العنصر الوجودي‬
x > 3 ‫اعطانا‬
x=4 ‫لو عوضنا‬
‫ صحيحه‬x > 4 ‫اذن‬
‫وبما ان عنصر واحد صحيح نقول‬
ᴲx P (x), is true.
Let Q(x) denote the statement "x = x + 1 ." What is the truth value of the quantification
ᴲx Q(x), where the domain consists of all real numbers?
‫هنا طلب مننا نعوض بجميع االرقام الحقيقه‬
x=x+1
‫ خطأ‬0 = 0 + 1 ‫لو قلنا‬
‫ولو عوضنا بجميع االرقام تكون جميعها خاطئه‬
ᴲx Q(x), is false ‫اذا نقول‬
.
Translating from English into Logical Expressions
Express the statement "Every student in this class has studied calculus" using
predicates and quantifiers?
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