MATHEMATICAL MATHEMATICAL REASONINGREASONING
STATEMENTSTATEMENT
A SENTENCE EITHER A SENTENCE EITHER TRUETRUE OR OR FALSE FALSE BUT NOT BOTH BUT NOT BOTH
STATEMENTSTATEMENT TEN IS LESS THAN ELEVEN TEN IS LESS THAN ELEVEN STATEMENTSTATEMENT ( TRUE ) ( TRUE )
TEN IS LESS THAN ONE TEN IS LESS THAN ONE STATEMENTSTATEMENT ( FALSE) ( FALSE)
PLEASE KEEP QUIET IN THE LIBRARYPLEASE KEEP QUIET IN THE LIBRARY NOT A STATEMENTNOT A STATEMENT
nono SentenceSentence statemestatementnt
NotNotstatemenstatementt
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
QUANTIFIERSQUANTIFIERS USED TO INDICATE THE QUANTITYUSED TO INDICATE THE QUANTITY ALLALL – TO SHOW THAT – TO SHOW THAT EVERY OBJECTEVERY OBJECT
SATISFIES CERTAIN CONDITIONSSATISFIES CERTAIN CONDITIONS
SOMESOME – TO SHOW THAT – TO SHOW THAT ONE OR ONE OR MOREMORE OBJECTS SATISFY CERTAIN OBJECTS SATISFY CERTAIN CONDITIONSCONDITIONS
QUANTIFIERSQUANTIFIERS
EXAMPLE :EXAMPLE : - All cats have four legsAll cats have four legs- Some even numbers are divisible by Some even numbers are divisible by
44- All perfect squares are more than 0All perfect squares are more than 0
OPERATIONS ON SETSOPERATIONS ON SETS
NEGATION NEGATION The truth value of a statement can be The truth value of a statement can be
changed by adding the word “changed by adding the word “notnot” ” into a statement.into a statement.
TRUE FALSE TRUE FALSE
NEGATIONNEGATION
EXAMPLEEXAMPLE
P : 2 IS AN EVEN NUMBER ( TRUE )P : 2 IS AN EVEN NUMBER ( TRUE )P (NOT P ) : P (NOT P ) : 2 IS NOT AN EVEN 2 IS NOT AN EVEN
NUMBER (FALSE NUMBER (FALSE ))
COMPOUND COMPOUND STATEMENTSTATEMENT
COMPOUND STATEMENTCOMPOUND STATEMENT
A compound statement is formed A compound statement is formed when two statements are combined when two statements are combined by using by using
“ “Or”Or” “ “and”and”
COMPOUND STATEMENTCOMPOUND STATEMENT
PP QQ P AND QP AND QTRUETRUE TRUETRUE TRUETRUETRUETRUE FALSEFALSE FALSEFALSEFALSEFALSE TRUETRUE FALSEFALSEFALSEFALSE FALSEFALSE FALSEFALSE
COMPOUND STATEMENTCOMPOUND STATEMENT
PP QQ P OR Q P OR Q TRUETRUE TRUETRUE TRUETRUETRUETRUE FALSEFALSE TRUETRUEFALSEFALSE TRUETRUE TRUETRUEFALSEFALSE FALSEFALSE FALSEFALSE
COMPOUND STATEMENTCOMPOUND STATEMENT
EXAMPLE :EXAMPLE :
P : All even numbers can be divided by 2 P : All even numbers can be divided by 2 ( TRUE )( TRUE )Q : -6 > -1Q : -6 > -1 ( FALSE )( FALSE ) P P andand Q : Q : FALSEFALSE
COMPOUND STATEMENTCOMPOUND STATEMENT
P : All even numbers can be divided by P : All even numbers can be divided by 2 2
( TRUE )( TRUE )Q : -6 > -1Q : -6 > -1 ( FALSE )( FALSE )
P P OROR Q : Q : TRUETRUE
IMPLICATIONSIMPLICATIONS SENTENCES IN THE FORMSENTENCES IN THE FORM
‘ ‘ IfIf pp thenthen q q ’ ,’ , wherewhere pp and and qq are statements are statements
And And p is the antecedentp is the antecedent
q is the consequentq is the consequent
IMPLICATIONSIMPLICATIONS
Example :Example :
If If xx33 = 64 = 64 , then , then x = 4x = 4 Antecedent : Antecedent : xx33 = 64 = 64 Consequent : Consequent : x = 4x = 4
IMPLICATIONSIMPLICATIONS
Example :Example : Identify the antecedent and consequent for the Identify the antecedent and consequent for the
implication below.implication below.
“ “ If the whether is fine this evening, If the whether is fine this evening, then I will play football”then I will play football”
Answer :Answer : Antecedent : the whether is fine this eveningAntecedent : the whether is fine this evening Consequent : I will play footballConsequent : I will play football
““pp if and only if if and only if qq””
The sentence in the form “The sentence in the form “pp if and if and only ifonly if qq” , is a compound statement ” , is a compound statement containing containing twotwo implications: implications:
a) If a) If pp , , then then qq b) If b) If qq , then , then pp
““pp if and only if if and only if qq””
“ “pp if and only if if and only if qq””
If p , then q If p , then q If q , then p
Homework !!!!Homework !!!! Pg: 96 No 1 and 2Pg: 96 No 1 and 2
Pg: 98 No 1, 2 ( b, c )Pg: 98 No 1, 2 ( b, c ) 4 ( a, b, c, d)4 ( a, b, c, d)
IMPLICATIONSIMPLICATIONS
The converse ofThe converse of “ “If p ,then q” If p ,then q” is is “ “if q , then p”.if q , then p”.
IMPLICATIONSIMPLICATIONS
Example :Example : If x = -5 , then 2x – 7 = -17If x = -5 , then 2x – 7 = -17
Mathematical reasoningMathematical reasoning
ArgumentsArguments
ARGUMENTS ARGUMENTS
What is argument ?What is argument ?- A process of making conclusion A process of making conclusion
based on a set of relevant based on a set of relevant information.information.
- Simple arguments are made up of Simple arguments are made up of two premises and a conclusiontwo premises and a conclusion
ARGUMENTSARGUMENTS
Example :Example : All quadrilaterals have four sides. A All quadrilaterals have four sides. A
rhombus is a quadrilateral. rhombus is a quadrilateral. Therefore, a rhombus has four sides.Therefore, a rhombus has four sides.
ARGUMENTSARGUMENTS
There are There are threethree forms of forms of arguments :arguments :
Argument Form I ( SyllogismArgument Form I ( Syllogism ) )Premise 1Premise 1 : All A are B : All A are B
Premise 2 : C is APremise 2 : C is A
Conclusion : C is BConclusion : C is B
ARGUMENTSARGUMENTSArgument Form 1( SyllogismArgument Form 1( Syllogism ) ) Make a conclusion based on the premises given Make a conclusion based on the premises given
below:below: Premise 1 : All even numbers can be divided Premise 1 : All even numbers can be divided
by 2 by 2 Premise 2 : 78 is an even numberPremise 2 : 78 is an even number
Conclusion : 78 can be divided by 2Conclusion : 78 can be divided by 2
ARGUMENTSARGUMENTS
Argument Form II ( Modus Argument Form II ( Modus Ponens ):Ponens ):
Premise 1Premise 1 : If : If pp , then , then qqPremise 2Premise 2 : p is true : p is true ConclusionConclusion : q is true : q is true
ARGUMENTSARGUMENTS
Example Example
Premise 1Premise 1 : If x = 6 , then x + 4 = 10 : If x = 6 , then x + 4 = 10Premise 2Premise 2 : x = 6 : x = 6ConclusionConclusion : x + 4 = 10 : x + 4 = 10
ARGUMENTSARGUMENTS
Argument Form III (Modus Argument Form III (Modus Tollens )Tollens )
Premise 1Premise 1 : If p , then q : If p , then qPremise 2Premise 2 : Not q is true : Not q is trueConclusionConclusion : Not p is true : Not p is true
ARGUMENTSARGUMENTS
Example : Example : Premise 1Premise 1 : If ABCD is a square, then : If ABCD is a square, then
ABCD ABCD has four sides has four sidesPremise 2Premise 2 : ABCD does not have four : ABCD does not have four
sides.sides.ConclusionConclusion : ABCD is not a square : ABCD is not a square
ARGUMENTSARGUMENTS
Completing the argumentsCompleting the arguments
recognise the argument formrecognise the argument form
Complete the argument according to Complete the argument according to its formits form
ARGUMENTSARGUMENTS
Example Example Premise 1 : All triangles have a sum of Premise 1 : All triangles have a sum of
interior interior angles of 180 angles of 180Premise 2 : ___________________________Premise 2 : ___________________________Conclusion : PQR has a sum of interior Conclusion : PQR has a sum of interior
angles of 180 angles of 180
PQR is a triangle
Argument Form I
ARGUMENTSARGUMENTS
Premise 1 : If x - 6 = 10 , then x = 16Premise 1 : If x - 6 = 10 , then x = 16
Premise 2 :__________________________Premise 2 :__________________________
Conclusion : x = 16 Conclusion : x = 16 Argument Form II
x – 6 = 10
ARGUMENTSARGUMENTS
Premise 1 : __________________________Premise 1 : __________________________
Premise 2 : x is not an even numberPremise 2 : x is not an even number
Conclusion : x is not divisible by 2Conclusion : x is not divisible by 2Argument Form III
If x divisible by 2 , then x is an even number
ARGUMENTSARGUMENTS
Homework :Homework :Pg : 103 Ex 4.5 No 2,3,4,5Pg : 103 Ex 4.5 No 2,3,4,5
MATHEMATICAL MATHEMATICAL REASONINGREASONING
DEDUCTION DEDUCTION AND AND
INDUCTIONINDUCTION
REASONINGREASONING There are two ways of making There are two ways of making
conclusions through reasoning by conclusions through reasoning by a) Deductiona) Deduction b) Inductionb) Induction
DEDUCTIONDEDUCTION
IS A PROCESS OF MAKING A IS A PROCESS OF MAKING A SPECIFIC CONCLUSION BASED ON A SPECIFIC CONCLUSION BASED ON A GIVEN GENERAL STATEMENT GIVEN GENERAL STATEMENT
DEDUCTIONDEDUCTION
Example : Example :
All students in Form 4X are present All students in Form 4X are present today.today.
David is a student in Form 4X.David is a student in Form 4X.Conclusion : David is present todayConclusion : David is present today
general
Specific
INDUCTIONINDUCTION
A PROCESS OF MAKING A GENERAL A PROCESS OF MAKING A GENERAL CONCLUSION BASED ON SPECIFIC CONCLUSION BASED ON SPECIFIC
CASES.CASES.
INDUCTIONINDUCTION
INDUCTIONINDUCTION
Amy is a student in Form 4X. Amy likes PhysicsCarol is a student in Form 4X. Carol likes PhysicsElize is a student in Form 4X. Elize likes Physics…………………………………………………….. Conclusion : All students in Form 4X like Physics .
REASONINGREASONING
DeductionDeduction
InductionInduction
GENERAL SPECIFIC