mathematical reasoning
DESCRIPTION
TRANSCRIPT
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
nono SentenceSentence statemestatementnt
NotNot
statemenstatementt
reasonreason
11 123 is 123 is divisible by divisible by 33
22
33 X-2 X-2 ≥ 9≥ 944 Is 1 a prime Is 1 a prime
number?number?
55 All octagons All octagons have eight sideshave eight sides
543 22
true
false
Neither true or false
A question
true
QUANTIFIERS
USED TO INDICATE THE QUANTITYALL – 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
PP QQ P AND QP AND Q
TRUETRUE TRUETRUE TRUETRUE
TRUETRUE FALSEFALSE FALSEFALSE
FALSEFALSE TRUETRUE FALSEFALSE
FALSEFALSE FALSEFALSE FALSEFALSE
COMPOUND STATEMENT
PP QQ P OR Q P OR Q
TRUETRUE TRUETRUE TRUETRUE
TRUETRUE FALSEFALSE TRUETRUE
FALSEFALSE TRUETRUE TRUETRUE
FALSEFALSE FALSEFALSE FALSEFALSE
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
‘ If p then q ’ , where
p and q are statements
And 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 weather is fine this evening, then I will play football”
Answer : Antecedent : the weather 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 p , then q If q , then p
Homework !!!!
Pg: 96 No 1 and 2
Pg: 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
ARGUMENTS
Mathematical reasoning
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 is B
Premise 2 : C is A
Conclusion : C is B
ARGUMENTSArgument 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 qPremise 2 : p is true Conclusion : q is true
ARGUMENTS
Example
Premise 1 : If x = 6 , then x + 4 = 10Premise 2 : x = 6Conclusion : x + 4 = 10
ARGUMENTS
Argument Form III (Modus Tollens )Premise 1 : If p , then qPremise 2 : Not q is trueConclusion : Not p is true
ARGUMENTS
Example : Premise 1 : If ABCD is a square, then ABCD
has four sidesPremise 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 180Premise 2 : ___________________________Conclusion : PQR has a sum of interior
angles of 180
PQR is a triangle
Argument Form I
ARGUMENTS
Premise 1 : If x - 6 = 10 , then x = 16
Premise 2 :__________________________
Conclusion : x = 16
Argument Form II
x – 6 = 10
ARGUMENTS
Premise 1 : __________________________
Premise 2 : x is not an even number
Conclusion : x is not divisible by 2
Argument Form III
If x divisible by 2 , then x is an even number
ARGUMENTS
Homework :Pg : 103 Ex 4.5 No 2,3,4,5
DEDUCTION AND
INDUCTION
MATHEMATICAL REASONING
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 :
All students in Form 4X are present today.David is a student in Form 4X.Conclusion : David is present today
general
Specific
INDUCTION
A PROCESS OF MAKING A GENERAL CONCLUSION BASED ON SPECIFIC CASES.
INDUCTIONINDUCTION
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
Induction
GENERAL SPECIFIC