MATHEMATICAL REASONING MATHEMATICAL REASONING

STATEMENT A SENTENCE EITHER TRUE OR FALSE BUT NOT BOTH A SENTENCE EITHER TRUE OR FALSE BUT NOT BOTH

STATEMENT TEN IS LESS THAN ELEVEN TEN IS LESS THAN ELEVEN STATEMENT ( TRUE ) STATEMENT ( TRUE ) TEN IS LESS THAN ONE TEN IS LESS THAN ONE STATEMENT ( FALSE) STATEMENT ( FALSE) PLEASE KEEP QUIET IN THE LIBRARY PLEASE KEEP QUIET IN THE LIBRARY NOT A STATEMENT NOT A STATEMENT

noSentencestatementNotstatementreason is divisible by X-2 ≥ 9 4 Is 1 a prime number? 5 All octagons have eight sides true false Neither true or false A question true

QUANTIFIERS USED TO INDICATE THE QUANTITY USED TO INDICATE THE QUANTITY ALL – TO SHOW THAT EVERY OBJECT SATISFIES CERTAIN CONDITIONS ALL – TO SHOW THAT EVERY OBJECT SATISFIES CERTAIN CONDITIONS SOME – TO SHOW THAT ONE OR MORE OBJECTS SATISFY 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. The truth value of a statement can be changed by adding the word “ not ” into a statement. TRUE FALSE TRUE FALSE

NEGATION EXAMPLE P : 2 IS AN EVEN NUMBER ( TRUE ) P : 2 IS AN EVEN NUMBER ( TRUE )  P (NOT P ) : 2 IS NOT AN EVEN NUMBER (FALSE )

COMPOUND STATEMENT

A compound statement is formed when two statements are combined by using A compound statement is formed when two statements are combined by using “Or” “Or” “and” “and”

COMPOUND STATEMENT P Q P AND Q P AND Q TRUETRUE TRUE TRUEFALSE FALSE FALSETRUE FALSE FALSEFALSE FALSE

COMPOUND STATEMENT P Q P OR Q P OR Q TRUETRUE TRUE TRUEFALSE TRUE FALSETRUE TRUE FALSEFALSE FALSE

COMPOUND STATEMENT EXAMPLE : P : All even numbers can be divided by 2 ( TRUE ) ( TRUE ) Q : -6 > -1 ( FALSE ) ( FALSE ) P and Q : P and Q : FALSE FALSE

COMPOUND STATEMENT P : All even numbers can be divided by 2 ( TRUE ) ( TRUE ) Q : -6 > -1 ( FALSE ) ( FALSE ) P OR Q : P OR Q : TRUE TRUE

IMPLICATIONS SENTENCES IN THE FORM SENTENCES IN THE FORM ‘ If p then q ’, ‘ If p then q ’, where where p and q are statements p and q are statements And p is the antecedent q is the consequent q is the consequent

IMPLICATIONS Example : If x 3 = 64, then x = 4 If x 3 = 64, then x = 4 Antecedent : x 3 = 64 Antecedent : x 3 = 64 Consequent : x = 4 Consequent : x = 4

IMPLICATIONS Example : Example : Identify the antecedent and consequent for the implication below. Identify the antecedent and consequent for the implication below. “ If the whether is fine this evening, then I will play football” “ If the whether is fine this evening, then I will play football” Answer : Answer : Antecedent : the whether is fine this evening Antecedent : the whether is fine this evening Consequent : I will play football 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 a) If p, then q b) If q, then p b) If q, then p

“p if and only if q” “p if and only if q” “p if and only if q” If p, then q If q, then p

Homework !!!! Pg: 96 No 1 and 2 Pg: 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)

IMPLICATIONS The converse of The converse of “If p,then q” “If p,then q” is is “if q, then p”. “if q, then p”.

IMPLICATIONS Example : If x = -5, then 2x – 7 = -17

Mathematical reasoning Arguments

ARGUMENTS What is argument ? -A-A-A-A process of making conclusion based on a set of relevant information. - S- S- S- 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. All quadrilaterals have four sides. A rhombus is a quadrilateral. Therefore, a rhombus has four sides.

ARGUMENTS There are three forms of 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 1 : All even numbers can be divided by 2 Premise 2 : 78 is an even number 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 recognise the argument form Complete the argument according to its form Complete the argument according to its form

ARGUMENTS Example Premise 1 : All triangles have a sum of interior angles of 180  Premise 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

MATHEMATICAL REASONING DEDUCTIONANDINDUCTION

REASONING There are two ways of making conclusions through reasoning by There are two ways of making conclusions through reasoning by a) Deduction a) Deduction b) Induction b) Induction

DEDUCTION 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

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 A PROCESS OF MAKING A GENERAL CONCLUSION BASED ON SPECIFIC CASES. 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 Deduction Induction Induction GENERALSPECIFIC