BINARY OPERATIONS : BINARY OPERATIONS
My name is Ignatius George. I am from the Kottayam District of Kerala State, India. : My name is Ignatius George. I am from the Kottayam District of Kerala State, India. I have about 25 years of experience in the art of teaching mathematics in and abroad India.I was a Mathematics Examiner of the ICSE Examinations Council of India in 1992
BINARY OPERATIONS : BINARY OPERATIONS Binary operations
Commutative property
Associative Property
Existence of identity element
Existence of Inverse
1.NUMBER SYSTEM : 1.NUMBER SYSTEM
2.Number system and notations : 2.Number system and notations Real Nos= R = { -4 , 7 , 0.2 , 2/3 , 4 ¾ , 15 , 0 ,
1.333….. , , - 23 , 4 , 5.4 , , -3.7 }
Rational numbers = Q = { - 4 , 7 , 0.2 , 2/3 , 4 ¾, 15 , 0
, 1.333….. , - 23 , 4 , 5.4 , -3.7 }
Irrational nos { , }
Integers = Z = { - 4 , 7 , 15 , 0 , -23 , 4 }
Natural Nos = N = { 4 , 7 }
3.The symbol ‘*’ is used to represent some operations given below.State the name of the exact operation that stands for this symbol ‘ * ‘ used in each .(i) 3 * (-5 ) = (-2) in Z(ii) 16 * 8 = 2 in N(iii) 3 * 4 = 25 in N(iv) (2.5) * 4 = 10 in R(v) { 1 , 3 ,4 } * {3 , 4 , 7 } = {3 , 4} in P : 3.The symbol ‘*’ is used to represent some operations given below.State the name of the exact operation that stands for this symbol ‘ * ‘ used in each .(i) 3 * (-5 ) = (-2) in Z(ii) 16 * 8 = 2 in N(iii) 3 * 4 = 25 in N(iv) (2.5) * 4 = 10 in R(v) { 1 , 3 ,4 } * {3 , 4 , 7 } = {3 , 4} in P Ans: (i) addition (ii) division (iii) sum of squares (iv) multiplication (v) intersection
4. Example 1 for Binary Operations Let A = {-1 , 0 , 1}Consider f: (A X A) ? A such that f(a, b) = ab for all a, b elements of A : 4. Example 1 for Binary Operations Let A = {-1 , 0 , 1}Consider f: (A X A) ? A such that f(a, b) = ab for all a, b elements of A
5.Which of the following operations are binary operations ?(a) addition in the set of natural numbers (N)(b) subtraction in the set integers (Z)(c) subtraction in N(d) Multiplication in R(e) Addition in the set of odd numbers(f) division in R : 5.Which of the following operations are binary operations ?(a) addition in the set of natural numbers (N)(b) subtraction in the set integers (Z)(c) subtraction in N(d) Multiplication in R(e) Addition in the set of odd numbers(f) division in R Answer: (a) , (b) , (d)
6 Which of the following are binary operations ?(a) Addition of numbers in N , Z , Q , R(b) Subtraction of numbers in N , Z , Q , R© Multiplication of numbers in N , Z , Q , R(d) Division of numbers in N , Z , Q , R : 6 Which of the following are binary operations ?(a) Addition of numbers in N , Z , Q , R(b) Subtraction of numbers in N , Z , Q , R© Multiplication of numbers in N , Z , Q , R(d) Division of numbers in N , Z , Q , R Answers: (a) All (b) except N © All
(d) not in N and Z. it is a binary operation in
Q - {0} and R – {0}
I take online tuitions from grade 7 to grade 12 (CBSE /Cmbridge / Equivalent ) : I take online tuitions from grade 7 to grade 12 (CBSE /Cmbridge / Equivalent ) In case you need a mathematics tutor, you can contact me using any of the following means:
e- mail gntsgeorge@yahoo.co.in
y/Messenger : gntsgeorgexx@yahoo.in
skype: georgeignatius9
google talk: georgeignatiusxx
Tel: + 9104812790195
7.A binary operation in A is said to be COMMUTATIVE if ---for every ( a , b) A X B , a * b = b * a : 7.A binary operation in A is said to be COMMUTATIVE if ---for every ( a , b) A X B , a * b = b * a
8. Which of the following binary operations are not commutative ?(a) subtraction in Z , the set of Integers(b) multiplication in R , the set of real numbers© division in Q , the set of rational numbers. : 8. Which of the following binary operations are not commutative ?(a) subtraction in Z , the set of Integers(b) multiplication in R , the set of real numbers© division in Q , the set of rational numbers. Answer : a and c
9.Let * is a binary operation in N , described as “sum of the squares of”.Show that this binary operation is commutative : 9.Let * is a binary operation in N , described as “sum of the squares of”.Show that this binary operation is commutative Solution:
Let a and b be any two elements of N.
Then a * b = a2 + b2
= b2 + a2 = b * a
ie a * b = b * a
So: The given binary operation will be commutative.
10.Let * indicates a binary operation in N such that f (a , b) = lcm ( a , b ) where a and b are elements of N.Show that this binary operation is commutative. : 10.Let * indicates a binary operation in N such that f (a , b) = lcm ( a , b ) where a and b are elements of N.Show that this binary operation is commutative. Solution
Let a and b be any two natural numbers.
We know that lcm of a and b is the same as lcm of b and a
or lcm (a , b) = lcm (b , a)
or a * b = b * a
Hence the given binary operation is commutative.
I take online tuitions from grade 7 to grade 12 (CBSE /Cmbridge / Equivalent ) : I take online tuitions from grade 7 to grade 12 (CBSE /Cmbridge / Equivalent ) In case you need a mathematics tutor, you can contact me using any of the following means:
e- mail gntsgeorge@yahoo.co.in
y/Messenger : gntsgeorgexx@yahoo.in
skype: georgeignatius9
google talk: georgeignatiusxx
Tel: + 9104812790195
11.A Binary operation , * in A is said to be associative when-(a * b) * c = a * ( b * c)where a , b and c are elements of A : 11.A Binary operation , * in A is said to be associative when-(a * b) * c = a * ( b * c)where a , b and c are elements of A
12.Prove that the binary operation * defined in N as m * n = l. c. m, (m , n) , m , n N, is associative. : 12.Prove that the binary operation * defined in N as m * n = l. c. m, (m , n) , m , n N, is associative. Let a , b , c are any three elements of N
a * (b * c) = a * l.c.m ( b, c)
= l.c.m { a , l.c.m(b , c) }
= l.c.m ( a , b , c)
(a * b) * c = l.c.m (a , b) * c
= l.c.m{l.c.m(a,b) , c}
= l.c.m ( a , b , c )
Therefore a * (b * c) = (a * b) * c
Or the operation * is associative
Determine whether the binary operation defined in N as a * b = ab , a , b N is associative ? : Determine whether the binary operation defined in N as a * b = ab , a , b N is associative ? Let a , b and c be any three elements of N
a * (b * c ) = a * bc =
(a * b) * c = ab * c =
Hence, a*(b*c) is not equal to (a*b)*c since bc is not the same as bc.
Therefore the given operation is not associative
14. Given * is a binary operation defined on R by a * b = a + for a, b R. Determine whether ( 2 * 6) * 10 is equal to 2 * (6 * 10). Hence find out whether the operation is associative or not : 14. Given * is a binary operation defined on R by a * b = a + for a, b R. Determine whether ( 2 * 6) * 10 is equal to 2 * (6 * 10). Hence find out whether the operation is associative or not (2 * 6) = 2 + (6/2) = 2 + 3 = 5
(2 * 6) * 10 = 5*10 =5 + (10/2) = 5 + 5 = 10
6*10 = 6 + (10/2) = 6 + 5 = 11
So 2 * (6 * 10) = 2 * 11 = 2 + (11/2) = 7.5
Therefore they are not equal and hence the operation not associative.
15.True or false ?(a) identity element exists for the binary operation “addition of numbers” in N(b) identity element exists for the operation “multiplication of numbers” in N© identity element exists for the operation “ addition of numbers” in R(d) identity element exists for the binary operation “division of numbers” in R – {0} : 15.True or false ?(a) identity element exists for the binary operation “addition of numbers” in N(b) identity element exists for the operation “multiplication of numbers” in N© identity element exists for the operation “ addition of numbers” in R(d) identity element exists for the binary operation “division of numbers” in R – {0} Identity element exists in the cases , (b) , © and (d)
16.When a binary operation * : A X A ? A is given ,an element e in A , if it exists , is called an identity element for the operation * if a * e = a = e * a : 16.When a binary operation * : A X A ? A is given ,an element e in A , if it exists , is called an identity element for the operation * if a * e = a = e * a
I take online tuitions from grade 7 to grade 12 (CBSE /Cmbridge / Equivalent ) : I take online tuitions from grade 7 to grade 12 (CBSE /Cmbridge / Equivalent ) In case you need a mathematics tutor, you can contact me using any of the following means:
e- mail gntsgeorge@yahoo.co.in
y/Messenger : gntsgeorgexx@yahoo.in
skype: georgeignatius9
google talk: georgeignatiusxx
Tel: + 9104812790195
17.Find each of the following(i) inverse of 4 in the Binary operation, “addition” in R (ii) inverse of 7 in the Binary operation, multiplication in R(iii) identity element in the binary operation subtraction in Z (iv) inverse of -13 in the binary operation addition in Z (v) identity element in binary operation, division in Q - {0} (vi) inverse of 11 in the binary operation division in R – {0} : 17.Find each of the following(i) inverse of 4 in the Binary operation, “addition” in R (ii) inverse of 7 in the Binary operation, multiplication in R(iii) identity element in the binary operation subtraction in Z (iv) inverse of -13 in the binary operation addition in Z (v) identity element in binary operation, division in Q - {0} (vi) inverse of 11 in the binary operation division in R – {0} Ans: (i) - 4 (ii) 1/7 (iii) 0 (iv) 13 (v) 1 (vi) 11
18 Let * be an operation defined on the set of positive rational numbers by a * b= for all a , b Q + Show that(i) * is a binary operation on Q+(ii) * is commutative(iii) * is associative(iv) Find the identity element of this operation(v) what is the inverse of a Q+ : 18 Let * be an operation defined on the set of positive rational numbers by a * b= for all a , b Q + Show that(i) * is a binary operation on Q+(ii) * is commutative(iii) * is associative(iv) Find the identity element of this operation(v) what is the inverse of a Q+