Download COMPOUND STATEMENT

MATHEMATICAL
REASONING
STATEMENT
A SENTENCE EITHER TRUE OR
FALSE BUT NOT BOTH
STATEMENT




TEN IS LESS THAN ELEVEN
STATEMENT ( TRUE )
TEN IS LESS THAN ONE
STATEMENT ( FALSE)
PLEASE KEEP QUIET IN THE LIBRARY
NOT A STATEMENT
Sentence
1
123 is
divisible by 3
true
2
3 4 5
false
3
X-2 ≥ 9
4
Is 1 a prime
number?
5
All octagons have
eight sides
2
2
statement Not
statement
reason
no
Neither true or false
A question
true
QUANTIFIERS



USED TO INDICATE THE QUANTITY
ALL – TO SHOW THAT EVERY OBJECT
SATISFIES CERTAIN CONDITIONS
SOME – TO SHOW THAT ONE OR MORE
OBJECTS SATISFY CERTAIN CONDITIONS
QUANTIFIERS
EXAMPLE :
-
-
All cats have four legs
Some even numbers are divisible by 4
All perfect squares are more than 0
OPERATIONS ON SETS
NEGATION
The truth value of a statement can be
changed by adding the word “not” into a
statement.
TRUE
FALSE
NEGATION
EXAMPLE
P : 2 IS AN EVEN NUMBER ( TRUE )
P (NOT P ) : 2 IS NOT AN EVEN
NUMBER (FALSE )
COMPOUND
STATEMENT
COMPOUND STATEMENT
A compound statement is formed when
two statements are combined by using


“Or”
“and”
COMPOUND STATEMENT
P
Q
P AND Q
TRUE
TRUE
TRUE
TRUE
FALSE
FALSE
FALSE
TRUE
FALSE
FALSE
FALSE
FALSE
COMPOUND STATEMENT
P
Q
P OR Q
TRUE
TRUE
TRUE
TRUE
FALSE
TRUE
FALSE
TRUE
TRUE
FALSE
FALSE
FALSE
COMPOUND STATEMENT
EXAMPLE :
P : All even numbers can be divided by 2
( TRUE )
Q : -6 > -1
( FALSE )
P
and Q
FALSE
:
COMPOUND STATEMENT
P : All even numbers can be divided by 2
( TRUE )
Q : -6 > -1
( FALSE )
P OR Q
TRUE
:
IMPLICATIONS

SENTENCES IN THE FORM
where
And
‘ If
p
then q ’ ,
p and q are statements
p is the antecedent
q is the consequent
IMPLICATIONS
Example :
If x3 = 64 , then x = 4
Antecedent : x3 = 64
Consequent : x = 4
IMPLICATIONS
Example :
Identify the antecedent and consequent for the
implication below.
“ If the whether is fine this evening, then I
will play football”
Answer :
Antecedent : the whether is fine this evening
Consequent : I will play football
“p if and only if q”
The sentence in the form “p if and only if
q” , is a compound statement containing
two implications:
a) If p , then q
b) If q , then p
“p if and only if q”
“p if and only if q”
If p , then q
If q , then p
Homework !!!!
 Pg:
 Pg:
96 No 1 and 2
98 No 1, 2 ( b, c )
4 ( a, b, c, d)
IMPLICATIONS
The converse of
“If p ,then q”
is
“if q , then p”.
IMPLICATIONS
Example
:
If x = -5 , then 2x – 7 = -17
Mathematical reasoning
Arguments
ARGUMENTS
What is argument ?
- A process of making conclusion based on
a set of relevant information.
-
Simple arguments are made up of two
premises and a conclusion
ARGUMENTS
Example :
All quadrilaterals have four sides. A
rhombus is a quadrilateral. Therefore, a
rhombus has four sides.
ARGUMENTS

There are three forms of
arguments :
Argument Form I ( Syllogism )
Premise 1 : All A are B
Premise 2
: C is A
Conclusion : C is B
ARGUMENTS
Argument Form 1( Syllogism )
Make a conclusion based on the premises given
below:
Premise 1 : All even numbers can be divided
by 2
Premise 2 : 78 is an even number
Conclusion
: 78 can be divided by 2
ARGUMENTS
Argument Form II ( Modus Ponens ):
Premise 1 : If p , then q
Premise 2 : p is true
Conclusion : q is true
ARGUMENTS
Example
Premise 1 : If x = 6 , then x + 4 = 10
Premise 2 : x = 6
Conclusion : x + 4 = 10
ARGUMENTS
Argument Form III (Modus Tollens )
Premise 1 : If p , then q
Premise 2 : Not q is true
Conclusion : Not p is true
ARGUMENTS
Example :
Premise 1 : If ABCD is a square, then
ABCD
has four sides
Premise 2 : ABCD does not have four
sides.
Conclusion : ABCD is not a square
ARGUMENTS
Completing the arguments


recognise the argument form
Complete the argument according to its
form
ARGUMENTS
Example
Premise 1 : All triangles have a sum of
interior
angles of 180
PQR is a triangle
Premise 2 :
___________________________
Conclusion : PQR has a sum of interior
Argument
Form I
angles
of 180
ARGUMENTS
Premise 1 : If x - 6 = 10 , then x = 16
x – 6 = 10
Premise 2
:__________________________
Conclusion : x Argument
= 16 Form II
ARGUMENTS
Premise 1 : If x divisible by 2 , then x is an even
number
__________________________
Premise 2 : x is not an even number
Conclusion : Argument
x is not Form
divisible
by 2
III
ARGUMENTS
Homework :
Pg : 103 Ex 4.5 No 2,3,4,5
MATHEMATICAL
REASONING
DEDUCTION
AND
INDUCTION
REASONING

There are two ways of making conclusions
through reasoning by
a) Deduction
b) Induction
DEDUCTION
IS A PROCESS OF MAKING A
SPECIFIC CONCLUSION BASED ON A
GIVEN GENERAL STATEMENT
DEDUCTION
Example :
general
All students in Form 4X are present today.
David is a student in Form 4X.
Conclusion : David is present today
Specific
INDUCTION
A PROCESS OF MAKING A GENERAL
CONCLUSION BASED ON SPECIFIC
CASES.
INDUCTION
INDUCTION
Amy is a student in Form 4X. Amy likes
Physics
Carol is a student in Form 4X. Carol likes
Physics
Elize is a student in Form 4X. Elize likes
Physics
……………………………………………………..
Conclusion : All students in Form 4X like
Physics .
REASONING
Deduction
GENERAL
SPECIFIC
Induction