Kendriya Vidyalaya IFFCO,Gandhidham Maths Project On Binary Operations

1. Kendriya Vidyalaya IFFCO,GandhidhamMaths ProjectOn Binary Operations

2. DETAILS • Name: Swetha,Sumouly,Prachi,Sejal. • Standard: XII • Subject: Mathematics • Topic: Binary Operations • Teacher-In-charge: Geetha Ma’am • School: KV,IFFCO Gandhidham. • Year: 2010-2011

3. ACKNOWLEDGEMENT We hereby extend our sincere gratitude to our teacher/mentor Madam Geetha for alloting such an interesting project,to our Principal Mrs.Sangeeta Gutain , for her co-operation and guidance,to our parents for their help,concern and blessings.We gratefully acknowledge the valuable and precious contributions and support of the above mentioned people in making this project a SUCCESS………..

4. CONTENTS

5. INTRODUCTION • BINARY OPERATION: A binary operation * on a set A is a function from AXA→A. We denote it by *(a,b)=a*b.+,-,*,/,these operations between two operands result in binary operation.

6. E.g. A binary operation on ‘R’ +:R×R→R is given by (a,b)→a+b E.g. A binary operation of division is not possible on ‘R’ because ‘R’ includes 0 also and in ‘a/b’ form if ‘b’ is zero then product becomes not defined.

7. COMMUTATIVE BINARY OPERATION • COMMUTATIVE OPERATION A binary operation * on the set x is called commutative if a*b=b*a for all a,b ε X. E.g.5+2=2+5, where a=5,b=2 and * is operation (+), this satisfies the condition of a+b=b+a,i.e 7=7.

8. E.g.Show that * is a function from RXR→R defined by a*b=a+2b is not commutative? Solution: a*b=a+2b (given) b*a=b+2a, but it is clear that a*b ≠ b*a , that means a+2b ≠ b+2a. let us take a=2 and b=3 a+2b= 2+2(3)=8,but b+2a=3+2(2)=7, it is clear that a+2b ≠ b+2a as 8 ≠ 7. Hence proved…

9. ASSOCIATIVE BINARY OPERATION • ASSOCIATIVE OPERATION: A binary operation * : AXA →A is said to be associative if (a*b)*c=a*(b*c) for all a,b,c ε A. E.g. (8+5)+2=8+(5+2),a=8,b=5 & c=2 and * is operation(+),this satisfies the condition (a*b)*c=a*(b*c) implies 15=15.

10. E.g.Show that * is a function from RXR→R defined by a*b=a+2b is not associative? Solution: (a*b)*c=(a+2b)*c (given) a*(b*c)=a*(b+2c) but it is clear that (a*b)*c ≠ a*(b*c) that means (a+2b)+c ≠ b+2a. let us take a=8 ,b=5 and c=3 (a+2b)*c= (8+2(5))*3=24, but a+(b+2c)=8*(5+2(3))=30 it is clear that (a+2b)*c ≠ a*(b+2c) as 24 ≠ 30. Hence proved…

11. IDENTITY ON A BINARYOPERATION • IDENTITY OPERATION: A binary operation * : AXA →A ,an element eεA if it exists, is called identity for operation *,if a*e=a=e*a for all aεA. • ADDITIVE IDENTITY: zero is identity for the addition operation. • MULTIPLICATIVE IDENTITY: one is identity for the multiplication operation.

12. E.g.Show that zero and one are additive and multiplicative identity on R. But there is no identity element for subtraction and division. Solution: a+0=0+a= a, shows 0 is additive identity and ax1=1xa= a , shows 1 is multiplicative identity. but a-e=e-a= a , there is no such value for e. a/e=e/a=a , there is no such value for e. Hence proved…

13. INVERTIBILITY OF A BINARY OPERATION • INVERTIBLE BINARY OPERATION: A binary operation * : AXA →A ,with the identity element in A , an element a ε A is said to be invertible with respect to the operation*, if there exists an element b in A such that a*b=e=b*a and b is called the inverse of a and is denoted by a-1.

14. E.g.Show that –a and 1/a(a!=0) are the inverse of a for addition(+) and multiplication operation(x) on R. Solution: As a+(-a)=a-a=0 & (-a)+a=0,shows –a is the inverse of a for addition. Similarly, for a!=0, a x (1/a) = 1 & (1/a) x a = 1, shows 1/a is the inverse of a for multiplication. Hence proved…

15. BIBLIOGRAPHY NCERT MATHEMATICS TEXTBOOK FOR CLASS XII… PART I

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