4 Statements

8/9/2019 4 Statements

1/22

4 Statements

(A) Determine Whether a Given Sentence is a Statement

1. A statement is a sentence which is either true or false but not both.

2. Sentences which are questions, instructions and exclamations are not statements.

Example 1:Determine whether the following sentences are statements or not. Give a

reason for your answer.(a) ! " #(b) A $entagon has % sides.(c) &s ' divisible by (d) *ind the $erimeter of a square with each side of 'cm.(e) +el$

Solution:(a) Statement- it is a false statement.(b) Statement- it is a true statement.(c) ot a statement- it is a question.(d) ot a statement- it is an instruction.(e) ot a statement- it is an exclamation.

(B) Determine Whether a Statement is True or False

Example 2:Determine whether each of the following statements is true or false.(a) / is a $rime number (b) 01 2 0/(c) is a factor of #.

Solution:(a) 3rue(b) *alse(c) *alse


2/22

(!) !onstructin" Statements #sin" $um%ers an& S'm%ols

1. 3rue and false statements can be constructed with numbers and mathematical symbols.

Example :4onstruct (i) a true statement, (ii) a false statement,using the following numbers and mathematical symbols.(a) 5, ', #, x, "(b) 6a, b, c7, 6d7 ,υ =

Solution:(a)(i) A true statement8 5 x ' " #(a)(ii) A false statement8 5 x # " '(b)(i) A true statement8 6d7υ 6a, b, c7 " 6a, b, c, d7(b)(ii) A false statement8 6d7υ 6a, b, c7 " 6d7

Statements Example (Sample Questions)

Example 1 8Determine whether the following sentences are statements or not. Give a reason for your answer.(a) ! " #(b) 9 : ' " %(c) A $entagon has % sides.(d) ' is a $rime number.(e) &s ' divisible by (f) *ind the $erimeter of a square with each side of 'cm.(g) +el$

Ans er 8(a) Statement- it is a false statement.(b) Statement- it is a true statement.(c) Statement- it is a true statement.(d) Statement- it is a false statement.(e) ot a statement- it is a question.(f) ot a statement- it is an instruction.


3/22

(g) ot a statement- it is an exclamation.

Quantifiers ‘All’ and ‘Some’*uanti+iers ,All- an& ,Some-

Statement usin" ,All- an& ,Some-

1 *uanti+iers are words that denote the number of ob;ects or casesreferred to in a given statement.

2 , =an' > and =ever' > describeeach an& ever' ob;ect orcase., =several > and =part o+ > describeone or moreob;ects or cases.

Example:4om$lete each of the following statements using the quantifiers =all> or =some>to ma?e the statement true.(a) @@@@@@@ $olygons have the same number of vertices and sides.(%) @@@@@@@ multi$les of 9 are even numbers.(c) @@@@@@@ of the whole numbers are divisible by /.(&) @@@@@@@ factors of ' are factors of 5 .

Solution:(a) All $olygons have the same number of vertices and sides.(%) Some multi$les of 9 are even numbers.(c) Some of the whole numbers are divisible by /.(&) All factors of ' are factors of 5 .

Quantifiers 'All' and 'Some' (Sample Questions)

Example 1 84om$lete each of the following statements using the quantifiers =all> or =some> toma?e the statement true.(a) @@@@@ rectangles are squares.(b) @@@@@ $rime numbers are odd numbers.(c) @@@@@ triangles have equal sides.(d) @@@@@ even numbers are divisible by 5.

Ans er 8(a) Some rectangles are squares.


4/22

(b) Some $rime numbers are odd numbers.(c) Some triangles have equal sides.(d) All even numbers are divisible by 5.

Example 2 84onstruct a true statement using the quantifier =all> or =some> for the given ob;ectand $ro$erty.

(a) b;ect8 multi$les of ' Bro$erty8 can be divided exactly by %(b) b;ect8 regular hexagon Bro$erty8 C equal sides(c) b;ect8 acute angles Bro$erty8 less than 9o

Ans er 8(a) Some multi$les of ' can be divided exactly by %.(b) All regular hexagons have C equal sides.(c) All acute angles are less than 9o .

Operations on Statements.perations on Statements

$a"atin" a Statement usin" ,$o- or ,$ot-

1 $e"ation of a statement refers tochan"in" thetruth value of the statement,that is,chan"in" a true statement to a +alse statement and vice versa, usingthe word,not- or ,no- .

Example 1:4hange the true value of the following statements by using =no> or =not>.(a) 1/ is a $rime number.(%) 9 is a multi$le of 9.

Solution:(a) 1/ is not a $rime number. (3rue to false)(%) 9not is a multi$le of 9. (*alse to true)

2 A compoun& statement can be formed bycom%inin" two given statementsusing the word,an&- .


5/22


6/22

Truth 0alues o+ !ompoun& Statements usin" ,An&-

/ hen two statements are combined using =an&>, atrue com$ound statementis obtained only if%oth statements aretrue .

&fone or %oth statements are+alse, then the com$ound statement is+alse.

3he truth ta%le 8Het p " statement 1 andq " statement 5.3he truth values for = p> and =q> are as follows8

p q p an& q (compoun& statement)3rue 3rue 3rue

3rue *alse *alse*alse 3rue *alse*alse *alse *alse

Example :Determine the truth value of the following statements.(a) 15 (: ) " : C and 1% : / " #.(%) % 2 and :' I :%.(c) +exagons have % sides and each of the interior angles is 9o .

Solution:(a)15 (: ) " : C E ( p is true )1% : / " # E (q is true )3herefore 15 (: ) " : C and 1% : / " # is atrue statement . (= p an& q- istrue )

(%)% 2 E ( p is true ) :' I :% E ( q is +alse )3herefore % 2 and :' I :% is a+alse statement . (= p an& q- is +alse )

(c)+exagons have % sides. E ( p is +alse )Jach of the interior angles of +exagon is 9o . E (q is +alse )3herefore +exagons have % sides and each of the interior angles is 9o is a+alsestatement . (= p an& q- is +alse )


7/22

Truth 0alues o+ !ompoun& Statements usin" ,.r-

1 hen two statements are combined using =or >, a+alse com$ound statement isobtained only if%oth statements are+alse.

2 f one or %oth statements aretrue , then the com$ound statement istrue .

3he truth ta%le 8Het p " statement 1 andq " statement 5.3he truth values for = p> and =q> are as follows8

p q p an& q (compoun& statement)3rue 3rue 3rue3rue *alse 3rue*alse 3rue 3rue*alse *alse *alse

Example :Determine the truth value of the following statements.(a) C is divisible by ' or 9.(%) % " 5% or ' " C'.(c) % ! / 2 1' or K9 " 5.

Solution:(a)C is divisible by ' E ( p is true )C is divisible by 9 E (q is +alse )3herefore, C is divisible by ' or 9 is atrue statement . (= p or q- is true )

(%)% " 5% E ( p is +alse )' " C' E ( q is true )3herefore, % " 5% or ' " C' is a true statement . (= p or q- is true )

(c)% ! / 2 1' E ( p is +alse )K9 " 5 E (q is +alse )

3herefore, % ! / 2 1' or K9 " 5 is a+alse statement . (= p or q- is +alse )


8/22

Operations on Statements (Sample Questions)

Example 1 8*orm a com$ound statement by combining two given statements using the word

=and>.(a) 15 " C / % " %(%) % is a $rime number. % is an odd number.(c) Fectangles have ' sides. Fectangles have ' vertices.

Solution:(a) 15 " C and / % " %(%) % is a $rime number and an odd number.(c) Fectangles have ' sides and ' vertices.

Example 2 8*orm a com$ound statement by combining two given statements using the word=or>.(a) 1C is a $erfect square. 1C is an even number.(%) ' 2 . 0% I 01

Solution:(a) 1C is a $erfect square or an even number.(%) ' 2 or 0% I 01.

Example 8Determine whether each of the following statements is true or false.(a) (0') " 015 and 1 ! C " 19(%) 1 ./ " / and 15 ! (0 ) " 1#

Solution:hen two statements are combined using =an&>, atrue com$ound statement is

obtained only if%oth statements aretrue .&fone or %oth statements are+alse, then the com$ound statement is+alse.

(a) Loth the statements = (0') " 015> and =1 ! C " 19> are true. 3herefore, the

statement = (0') " 015 and 1 ! C " 19> istrue .


9/22


10/22

Antece&ent an& !onse3uent o+ an 4mplication

1 *or two statements p andq, the sentence =i+ p5 then q> is calledanimplication .

2 p is called theantece&ent . q is called theconse3uent .

Example:&dentify the antecedent and consequent of the following im$lications.(a) &fm " 5, then 5m5 ! m " 1(b) If P∪Q=P, then Q⊂P

Solution:(a) Antecedent8m " 5 4onsequent8: 5m5 ! m " 1

(b)Antecedent:P∪Q=P on!e"#ent:Q⊂P

4mplications o+ the Form , p i+ an& onl' i+ q-

1 3wo im$lications =i+ p5 then q> and =i+q5 then p> can be written as = p i+an& onl' i+ q>.

2 Hi?ewise, two statements can be written from a statement in theform, p i+ an& onl' i+ q- as follows8 &m$lication 18 &f p, thenq.

&m$lication 58 &fq, then p.Example 1:Given that p: x ! 1 " # q: x " /4onstruct a mathematical statement in the form of im$lication(a) &f p, thenq.(b) p if and only ifq.

Solution:


11/22

(a) &f x ! 1 " #, then x " /.(b) x ! 1 " # if and only if x " /.

Example 2:rite down two im$lications based on the following sentence8

x " C' if and only if x " '.

Solution:&f x " C', then x " '.&f x " ', then x " C'.

!onverse o+ an 4mplication

1 3he converse of an im$lication =if p, thenq> is =i+q5 then p>.

Example:State the converse of each of the following im$lications.(a) &f x5 ! x : 5 " , then ( x 0 1)( x ! 5) " .(b) &f x " /, then x ! 5 " 9.

Solution:(a) &f ( x 0 1)( x ! 5) " , then x5 ! x : 5.(b) &f x ! 5 " 9, then x " /.

Implications (Sample Questions)

Example 1 8rite down two im$lications based on the following statement.

y " 015% if and only if y " 0%.

Solution 8&m$lication 18 &f y " 015%, then y " 0%.&m$lication 58 &f y " 0%, then y " 015%.

Example 2 8rite down two im$lications based on the following statement.

# is a factor of 5' if and only if 5' can be divided exactly by #.


12/22

Solution 8&m$lication 18 # is a factor of 5' if 5' can be divided exactly by #.&m$lication 58 5' can be divided exactly by # if # is a factor of 5'.

Example 8State the converse of the following statement and hence, determine whether itsconverse is true or false.(a) &f 5 x 2 #, then x 2 '.(b) &f x is a multi$le of C, then it is a multi$le of .

Solution 8(a) 4onverse im$lication8 &f x 2 ', then 5 x 2 #. 3he converse is true.(b) 4onverse im$lication8 &f x is a multi$le of , then it is multi$le of C. 3he converse is false. (9 is a multi$le of but it is not a multi$le of C)

Argument(A) 6remises an& !onclusions

1 An argument is a $rocess of ma?ing conclusion based on several givenstatements.2 3he statements given are ?nown as $remises.

An argument consists of $remises and a conclusion.

Example 1:&dentify the $remises and conclusion of the following argument.(a) A $entagon has % sides. ABCDE is a $entagon. 3herefore, ABCDE has %sides.

Solution:6remise 1 8 A $entagon has % sides.6remise 2 8 ABCDE is a $entagon.!onclusion 8 ABCDE has % sides.

(B) Forms o+ Ar"uments


13/22


14/22

Ar"ument Form 444Bremise 18 &f p, thenq.Bremise 58 otq is true.4onclusion8 ot p is true.

Argument (Sample Questions)

Example 1 84om$lete the conclusion in the following argument8Bremise 18 All regular $olygons have equal sides.Bremise 58 ABCD is a regular $olygon.4onclusion8

Solution :4onclusion8 ABCD has equal sides.

Example 2 84om$lete the conclusion in the following argument8Bremise 18 &fm 2 ', then 5m 2 #.Bremise 58 5m I #4onclusion8

Solution :4onclusion8m I '.


15/22

Example 84om$lete the $remise in the following argument8

Bremise 18Bremise 58m$ n is not an even number.4onclusion8m andn are not even numbers.

Solution :Bremise 18 &fm andn are even numbers, thenm$ n is an even number.

Example / 84om$lete the $remise in the following argument8Bremise 18 &f x " , then x5 " 9.Bremise 584onclusion8 x %

Solution :Bremise 58 x5 % 9.

Deduction and InductionDe&uction an& 4n&uction

7easonin" %' De&uction an& 4n&uction

1 Feasoning by&e&uction is a $rocess of ma?ing aconclusion for aspeci+iccase based on a given"eneral statement .

2 Feasoning byin&uction is a $rocess of ma?ing a"enerali8ation based

on speci+ic cases .

9aths Tip

1 General statement N S$ecial conclusion NDe&uction2 S$ecific cases N General conclusion N4n&uction

Example: Determine whether the following conclusion is made based on a deductivereasoning or inductive reasoning.


16/22

(a) Area of triangle " O Lase +eight(i)

Area of P ABC " O /cm %cm" 1/.% cm5

(ii)

Area of P DEF " O /cm 'cm" 1' cm5

(%)1 " / (1)5 : C


17/22

55 " / (5)5 : C%/ " / ( )5 : C1 C " / (')5 : C / n5 : C,n " 1, 5, , 'Q

Solution:(a)3he speci+ic conclusion is made based on a"eneral statement Area oftriangle " O Lase +eight. 3herefore, the conclusion is made basedon&e&uctive reasonin" .(%) 3he "eneral conclusion / n5 : C,n " 1, 5, , 'Q is made based onspeci+iccases . 3herefore, the conclusion is based onin&uctive reasonin" .

Mathematics Reasoning Long Questions (Question 1 - 3)

*uestion 1 8

(%) 4om$lete the statement, in the answer s$ace, to form a true statement by using the quantifier =all> or =some>.

(c) rite down two im$lications based on the following statement8

A number is a $rime number if and only if itis only divisible by 1 and itself.

Solution:(a)(i) *alse(a)(ii) 3rue

(%) Some multi$les of are multi$les of C.

(c) &m$lication 18 &f a number is a $rime number, then it is only divisible by 1 and itself.


18/22

&m$lication 58 &f a number is only divisible by 1 and itself, then it is a $rime number.

*uestion 2 8(a) State if each of the following statements is true or false. (i) 5 " C or 5 ! " C (ii) 5 is a $rime number and % is an even number.

(%) rite down the converse of the following im$lication. +ence, state whether the converse is true or false.

&f x is a multi$le of 15,then x is a multi$le of .

(c) 4om$lete the $remise in the following argument8

Bremise 18 All hexagons have six sidesBremise 58 @@@@@@@@@@@@@@@@@@@@@

4onclusion8 ABCDEF has six sides.

Solution:(a)(i) 3rue(a)(ii) *alse

(%) 4onverse8 &f x is a multi$le of , then x is a multi$le of 15. 3he converse is+alse.

(c) Bremise 58 ABCDEF is a hexagon.

*uestion 8(a) 4om$lete each of the following statements with the quantifier =all> or =some> so that it will become atrue statement. (i) ;;;;;;;;;;; of the $rime numbers are odd numbers. (ii) ;;;;;;;;;;; $entagons have five sides.


19/22

(%) rite down two im$lications based on the following statement8

A R B " B if and only if A υ B " A.

(c) 4om$lete the $remise in the following argument8

Bremise 18 &f a number is a factor of 5', then it is a factor of '#.Bremise 58 15 is a factor of 5'.4onclusion8 @@@@@@@@@@@@@@@@@@@@@

Solution: (a)(i) Some of the $rime numbers are odd numbers. (a)(ii) All $entagons have five sides.

(%) &m$lication 18 &f A R B " B, then A υ B " A. &m$lication 58 &f A υ B " A, then A R B " B.

(c) 4onclusion8 15 is a factor of '#.

*uestion / 8(a) 4ombine the following two statements to form one true statement. Statement 18 (: )& " 9 Statement 58 : ( ) " 19

(%) 4om$lete the $remise in the following argument8

Bremise 18 @@@@@@@@@@@@@@@@@@@@@ Bremise 58 x is a multi$le of 5%.4onclusion8 x is a divisible of %.

(c) Ma?e a general conclusion by induction for the sequence of numbers /, 1',5/, Q which follows the following $attern.

< = (2) 1 > 1

1/ = (2)2

> 22< = (2) >


20/22

? = ???

Solution:(a) (: )& " 9 or : ( ) " 19.

(%) Bremise 18 All multi$les of 5% is divisible by %.

(c) (2) n > n 5 here n = 15 25 5 ?

*uestion 8(a) State if each of the following statements is true or false.

(i) 5 " # or ' " 1. . (ii) : C 2 : # and C 2 #.

Solution:(a)(i) True


21/22

(a)(ii) False


22/22

Documents

4 Statements