CONTRADICTION

1
class X
CONTRADICTION
CONJUNCTION
TAUTOLOGY
STATEMENT
EQUIVALENCE
IMPLICATION
MATHEMATIC LOGIC
OPEN SENTENCE
PROBLEMS
CONVERSE,INVERSE, CONTRAPOSITION
NEGATION
BIIMPLICATION
QUANTIFIER
DISJUNCTION
LOGICAL IMPLICATION
CONCLUSIONS
EXCERCISE
2
Statement is a sentence which is only right or
wrong, but can not right and wrong and the same
time.

• Example of Statement
• 3 is a prime number ? right statement
• 12 is an odd number ? wrong statement

STATEMENT

• Example of Non Statement
• What is your weight?
• For those who are sleepy, please sleep
• Chicken noodle is delicious

3
Open sentence is sentence which is contain
variable so the right value can not be determined.

• Example open sentence
• x y 10
• It is a solid thing

Open Sentence

• Open sentence can be changed into statement by
  changing the variable in universal set.
• Solution of open sentence is changed value in
  universal set which changes open sentence become
  right statement

4
From a statement can be made a new statement by
adding incorrect word in front of the first
statement or inserting not or no in the first
statement. The new statement can be found by
using the method called Negation marked with ?.
Example Negation of p 25 divisible by 2 is
p Incorrect if 25 divisible by 2 p 25
is not divisible by 2
NEGATION

• If statement p is true then ? p is false
• If statement p is false ? p is true

5
Disjunction is statement which is formed from
two statements p and q that is connected with
but and notated p ? q
A disjunction is false if two statements are false
DISJUNCTION
p q p ? q
T T T
T F T
F T T
F F F
TRUTH TABLE
Exercise
6
Conjunction is statement which is formed from
two statements p and q that is connected with
and/but/although/even though and notated p ? q
A conjunction is true if two statements are true.
CONJUNCTION
p q p ? q
T T T
T F F
F T F
F F F
TRUTH TABLE
EXERCISE
7
Implication is a compound statement which is
formed from two statements p and q, connected
with if then and notated p ? q
An implication is false if p is true and q is
false .
IMPLICATION
p q p ? q
T T T
T F F
F T T
F F T
TRUTH TABLE
EXERCISE
8
Bi-implication is a compound statement which is
formed from two statements p and q, connected
with if and only if and notated p ?q
A bi-implication is true if both statements have
the same truth.
BIIMPLICATION
p q p ? q
T T T
T F F
F T F
F F T
TRUTH TABLE
EXERCISE
9
Two statement A and B are equivalence, if have
the same truth, written A ? B

• These are some important equivalence
• Commutative Laws
• a. p ? q ? q ? p
• b. p ? q ? q ? p
• 2. Assosiative Laws
• a. p ? (q ? r) ? (p ? q) ? r
• b. p ? (q ? r) ? (p ? q) ? r

EQUIVALENCE
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3. Distributive Laws a. p ? (q ? r) ? (p ? q) ?
(p ? r) b. p ? (q ? r) ? (p ? q) ? (p ? r) 4.
de Morgan Laws a. (p ? q) ? p ? q b. (p ?
q) ? p ? q 5. Another equivalence
statements a. p ? q ? p ? q b. p ? q ? (p
? q) ? (q ? p) c. (p ? q) ? p ? q d. (p ?
q) ? (p ? q) ? (q ? p)
EQUIVALENCE
11
Tautology is a compound statements which is
always true for all true possibility of its
component statements.
Example Show that (p ? q) ?p is a tautology.
TAUTOLOGY
Answer
p q p ? q (p?q)?p
T T T T
T F F T
F T F T
F F F T
12
Contradiction is a compound statements which is
always false for all truth possibility of its
component statement.
Example Show that (p ? q) ? p is a
contradiction
CONTRADICTION
Answer
p q p ? q p (p?q) ? p
T T T F F
T F F F F
F T F T F
F F F T F
13
Implication Logic is a tautology which is
contain implication
Example Show that p ?(p ? q) is implication
logic.
LOGICAL IMPLICATION
Answer
p q p ? q p ? (p ? q)
B B B B
B S B B
S B B B
S S S B
14
From an implication p ? q can be formed another
implications are converse, inverse, and
contraposition. If p ? q is an implication
then q ? p is called converse ?p ? ?q is called
inverse ?q ? ?p is called contraposition
Converse Inverse Contraposition
15
Example Find converse, inverse and
contraposition from statement If ? is acute
angle, then cos ? is positif. Answer Converse
If cos ? is positive , then ? is acute
angle Inverse If ? is not acute angle, then
cos ? is not positive. Contraposition If cos
? is not positive, then ? is not acute angle
Converse Inverse Contraposition
Implication equal with contraposition p ? q ?
q ? p Inverse equal with converse p ?
q ? q ? p
16

• Universal Quantifier
• using all or each
• notated ?x ?S, p(x)
• read for all x so p(x)
• 2. Existential Quantifier
• using some or there are/is
• notated ?x ?S, p(x)
• read exist x so p(x)

QUANTIFIER
PROBLEMS
17

• Negation of Quantifier Statement
• (?x ?S, p(x)) ? ?x ?S, p(x)
• (? x ?S, p(x)) ? ? x ?S, p(x)
• Example
• Negation of All prime number is natural number
  is Exist a prime number is not natural number
• Negation of Exist plant which does not have
  leaves is All plant has leaves

QUANTIFIER
PROBLEMS
18

• Three basic conclusions , are modus ponens,
  modus tollens and syllogism
• Modus Ponens
• p ? q (premis)
• p (premis)
• ---------
• ? q (conclusion)

CONCLUSIONS
PROBLEMS
19
2. Modus Tollens p ? q (premis) ?q (premis) --
------- ? ?p (conclusion)
CONCLUSIONS
PROBLEMS
20
3. Syllogism p ? q (premis) q ?
r (premis) --------- ? p ? r (conclusion)
CONCLUSIONS
PROBLEMS
21
IMPLICATION
CONJUNCTION
CONCLUSIONS
PROBLEMS
BIIMPLICATION
EQUIVALENT
DISJUNCTION
22
DISJUNCTION EXERCISES
QUESTION NO. 1 Find the truth value form these
disjunction ! 3 is prime number or 3 is odd.
T
F
TO PROBLEMS MENU
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DISJUNCTION EXERCISES
QUESTION NO. 2 Find the truth value form these
disjunction ! sin2x cos2x 1 or sin 60?
½
T
F
TO PROBLEMS MENU
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DISJUNCTION EXERCISES
QUESTION NO. 3 Find the truth value form these
disjunction ! 0 is natural number or ½ is a
whole number.
T
F
TO PROBLEMS MENU
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DISJUNCTION EXERCISES
QUESTION NO. 4 Find the truth value form these
disjunction ! 13 divisible by 2 or 7 is prime
number.
T
F
TO PROBLEMS MENU
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CONJUNCTION EXERCISES

• Find the truth value form these conjunction !
• sin2x 1 cos2x and cos 60? ½
• 2 is prime number even tough 2 is even.
• - 3 is whole number but 3 is greater than 0.
• ?2 ? 8 ?10 and log 0 1

TO PROBLEMS MENU
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IMPLICATION EXERCISES

• Find the truth value form these implications!
• a. If 3 is factor of 12, then 12 is divisible by
  2.
• b. If log 10 1, then log 40 4.
• c. If 6 is prime number, then 6 is even.
• d. If x2 lt 0, then x2 1 gt 0
• e. If 22 x 23 26, then ?2 x ?3 ?6
• f. If sin 90? 0, then cos 90? 1
• g. If x2 gt 0, then ?4 ? 2

Next problems
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IMPLICATION EXERCISES
2. Known these statements have true value p
Farhan passes the exam q Farhan is happy Find
the truth value from each implication a. If
Farhan passes the exam, then he is happy b. If
Farhan passes the exam, then he is unhappy c. If
Farhan failed the exam, then he is happy d. If
Farhan failed the exam, then he is unhappy
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BI-IMPLICATION EXERCISES
1. Find the truth value form these bi-implication
! a. 0 is whole number if and only if 0 is
natural number b. 2 is prime number if and only
if 2 is factor of 12 c. tan 30? ?3 if and
only if cos180? 1 d. log 8 log 2 log 6 if
and only if log 10 1 e. 22 x 23 46 if and
only if ?2 ?3 ?5 f. x 1 0 if and only
if x2 0 g. x2 4 lt 0 if and only if -2 lt x lt 2
Next Problems
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BI-IMPLICATION EXERCISES
2. Known these statements have true value p
Olga is an actress q Olga is beautiful Find
the truth value from these bi-implication a. If
Olga is an actress, then she is beautiful b. If
Olga is an actress, then she is not beautiful c.
If Olga is not an actress, then she is
beautiful d. If Olga is not an actress, then she
is not beautiful
TO PROBLEMS MENU
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EQUIVALENCE EXERCISES
Find converse, inverse, and contraposition form
these implication 1. (p?q) ? r 2. (p?q) ?
r 3. p ? (q ?r) 4. If Tukul is an entertainer,
then he is funny 5. If ? ABC is equilateral,
then ? ABC is isosceles 6. If father goes to
his office, then he rides his car or motorcycle
TO PROBLEMS MENU
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CONCLUSION EXERCISES
Find valid or invalid of each argument 1. If
rainy then, Olga brings an umbrella Today is
raining ----------------------------------------
-------------------------- ? Olga is bringing
an umbrella 2. If Michael Jordan is a
basketball player, then he is tall Michael
Jordan is a basketball player ------------------
------------------------------------------------
? Michael Jordan is tall
Next Exercise
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CONCLUSION EXERCISES
3. If Zahra is passenger attendance, then she is
beautiful If Zahra is beautiful, then many
people like her
-------------------------------------------------
----------------------- ? If Zahra is passenger
attendance, then many people like her 4. If
there is sugar, then there is ants There is no
ants -------------------------------------------
----------------------- ? There is no sugar
TO PROBLEMS MENU
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DISJUNCTION EXERCISES
QUESTION NO. 1 Find the truth value from these
disjunction 3 is prime number or 3 is odd.
T
F
T
QUESTION NO. 2
TO PROBLEM MENU
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DISJUNCTION EXERCISES
QUESTION NO. 1 Find the truth value from these
disjunction 3 is prime number or 3 is odd.
T
F
T
QUESTION NO. 2
TO PROBLEMS MENU
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DISJUNCTION EXERCISES
QUESTON NO. 2 Find the truth value from these
disjunction ! sin2x cos2x 1 or sin 60? ½
T
F
T
QUESTION NO. 3
TO PROBLEM MENU
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DISJUNCTION EXERCISES
QUESTION NO. 2 Find the truth value from these
disjunction ! cos 60? ½ or sin 60? ½
T
T
F
QUESTION NO. 3
TO PROBLEMS MENU
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DISJUNCTION EXERCISES
QUESTION NO. 3 Find the truth value from these
disjunction ! 0 is natural number or ½ is whole
number.
S
F
T
QUESTION NO. 4
TO PROBLEMS MENU
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DISJUNCTION EXERCISES
QUESTION NO. 3 Find the truth value from each
disjunction! 0 is natural number or ½ is whole
number.
F
T
S
QUESTION NO. 4
TO PROBLEMS MENU
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DISJUNCTION EXERCISES
QUESTION NO. 4 Find the truth value from these
disjunction ! 13 is divisible by 2 or 7 is prime
number.
T
T
F
TO PROBLEMS MENU
41
DISJUNCTION EXERCISES
QUESTION NO. 4 Find the truth value from these
disjunction ! 13 is divisible by 2 or 7 is prime
number.
T
T
F
TO PROBLEMS MENU
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Good ,. You right!
43
Excellent,. Y ou right!
44
Well done ..!
45
Sorry..you are wrong!