1 / 15

Algebraic Structure Chapter -6

Algebraic Structure Chapter -6. MATHEMATICS CLASS - 9. Module Objectives. Recall different sets of numbers . Verify the closure property on a given set with respect to a given operation . Define binary operation .

dugan
Télécharger la présentation

Algebraic Structure Chapter -6

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Algebraic StructureChapter-6 MATHEMATICS CLASS - 9

  2. Module Objectives • Recalldifferent sets of numbers. • Verify the closureproperty on a given set with respect to a givenoperation. • Definebinaryoperation. • Check whether the givenoperation on a given set is a binaryoperation. • Define an algebraic structure. • Examine whether the given set with respect to a givenoperationis an algebraic structure.

  3. Introduction Let us have a recall on the different sets of numbersthatwe have studiedbefore. • Set of Natural NumbersN = {1,2,3,……} • Set of WholeNumbersW = {0,1,2,3,…} • Set of Integers Z = {……,-2 ,-1,0,+1,+2…….} • Set of Rational Numbers Q = p/q ,p,qЄ Z and q ≠ 0 • Set of Irrational Numbers Ir= • Set of Real Numbers R = { All rational and irrational numbers} Some of the symbols that we use to represent the properties on the above set of numbers are:

  4. Specialproperties on Sets of Numbers • Example 1 : Consider a set A = { -1,0,+1} This is a non-emptyset,theelements are -1,0,+1. Let us perform multiplication on thisset.All the possible products are givenbelow: Observe all the products.Do they belong to the same set A? All products are elements of set A Therefore,we say that A is closed under the operation –Multiplication This leads us to the following property: CLOSURE PROPERTY : In general,a,bЄ A Then a × b = c Є A with respect to the multiplication operation

  5. Specialproperties on Sets of Numbers • Example-2 : Consider set N = {1,2,3,…} Operation : + Let us perform the addition operation on the elements of set N. We observe that all the above sums are elements of set N. Therefore,set N is closed under additional operation. Note: Since this set N is infinite the summation property is common for all ,we come to a conclusion that sum of two natural numbers is also natural CLOSURE PROPERTY : In general,a,bЄ N, then ,a + b = c Є N with respect to the addition operation

  6. Specialproperties on Sets of Numbers • Example -3 : Given set : Z Operation : × Z = {……,-2 ,-1,0,+1,+2…….} Let us perform multiplication on Z. Observe that the product is either +ve integer, -ve integer or zero. Hence, in general, a,bЄ Z Then a × b = c Є Z Hence,Z satisfies the closure property under multiplication operation. CLOSURE PROPERTY : In general,a,bЄ Z ,then a × b = c Є Z with respect to the multiplication operation

  7. Specialproperties on Sets of Numbers Example-4 : Given set : S,Set of all evennaturalnumbers. Operation : + Let us perform addition operation on the set S = {2,4,6,8,10…} i.e.: 2 + 4 = 6 Є S 2 + 6 = 8 Є S 4 + 6 = 10 Є S Thus,the sum of any two even natural numbers is also an even natural number. In general, a,b, Є S , a+b = c Є S . Therefore,S satisfies the closure property with respect to addition. Example-5 : Given set : S,Set of all evennaturalnumbers. Operation : - Let us performsubtractionoperation on the set S = {2,4,6,8,10…} i.e.: 2 - 4 = -2 S 4 - 2 = 2 Є S 4 - 6 = -2  S In general , a,b, Є S , a- b = c  S . Therefore,S does not satisfy the closure property with respect to subtraction.

  8. Specialproperties on Sets of Numbers Example-6 : Given set : P,Set of all oddnaturalnumbers. Operation : + Let us perform addition operation on the set P = {1,3,5,7,9…} i.e: 1 + 3 = 4  P 1 + 5 = 6  P 3 + 5 = 8  P 5 + 7 = 12  P Thus,the sum of any two odd natural numbers is always not an odd natural number. In general , for all a,b, Є P , a+b = c  P Therefore,S does not satisfy the closure property with respect to addition. Example - 7: Given set : P, Set of all oddnaturalnumbers. Operation : × P = {……1,3,5,7…….} Let us perform multiplication on P. i.e: 1 × 3 = 3 Є P 3 × 5 = 15 Є P 3 × 7 = 21 Є P In general ,for all a,b, Є P, a x b = c Є P . Therefore P satisfies the closure property with respect to multiplication operation.

  9. Overview on the Closureproperties

  10. Binary Operations • In examples 1,2,3,4 and 7 , the given sets are satisfyingclosurepropertywith respect to the givenoperations. Suchoperations are calledBinary Operations. • Note : 1) `*’ isanyoperationbesides the four fundamentaloperations. 2) `a’ and `b’ need not benumbers.Theycanalsobe matrices or vectors. Binary Operations : The operations for which a given set satisfies the closure property. It is denoted as *.If a and b are any two elements of a set S and if a * b = c ЄS,then * is a Binary operation.

  11. Algebraic Structure The relationship between closure property and binary operation can be stated as follows: Any set which satisfies the Closure Property => The operation used on that set is a binary operation Any operation is a binary operation on a set => That set satisfies closure property under that operation ALGEBRAIC STRUCTURE: A non – empty set S equipped with certain binary operation is called an algebraic structure and is denoted by the pair (S,*) Note: The word structure is used to convey the presence of some order or a pattern. Let us solve some problems with binary operations and algebraic structure.!

  12. Algebraic Structures(Examples) Example-1 : Check whether * is a Binary operation on N , if * is defined by a * b = a + b + 1 a,bЄ N Solution : Set N = {1,2,3,…} Operation: * is defined by a * b = a+ b + 1 a,bЄ N To verify : Substitute a = 3, b = 4 in a* b = a + b + 1 => 3 * 4 = 3 + 4 + 1 = 8 and 8 Є N Therefore , the result will be a natural number and a * b = a + b + 1 Є N a,bЄ N Hence N is closed under * and * is a Binary operation on N. Example-2: Check whether * is a Binary operation on N , if * is defined by a * b = aba,bЄ N Solution: Given set: N Operation : * and is defined by a * b = aba,bЄ N To Verify : Substitute a = 3 , b = 2 in a * b = ab => 3 * 2 = 32 = 9 and 9 Є N Since the square of any two natural numbers is also a natural number, a * b = ab Є N a ,b Є N Hence N is closed under * and * is a Binary operation on N.

  13. Algebraic Structures(Examples) Example 3 : Given Set : Z Operation : * and is defined by a * b = (a+b)/2 a,bЄ Z Verify whether (Z,*) is an algebraic structure? Solution : Since * is defined by a* b = (a+b) / 2 a,bЄ Z Substitute the values , a = 1,b = 2 in a * b = (a+b)/2 1 * 2 = (1+2) / 2 1 * 2 = 3/2 But 3/2  Z Hence Z is not closed under * and * is not a Binary operation. So, (Z,*) is not an Algebraic Structure. Example 4 : Verify whether (Q,*) is an algebraic structure if * is defined by a * b = a 2 + b2 , where a,bЄ Q Solution : Substitute a = ½ , b = 3/2 in a * b = a 2 + b2 ½ * 3/2 = (1/2)2 + (3/2) 2 = (1/4) + (9/4) = (10/4) = 5/2 Є Q Hence, Q is closed under the binary operation * and (Q,*) is a Algebraic Structure.

  14. Do ityourself! 1 . Which of the following is a Binary Operation on the given set ? a) a* b = a + b + 2, a,bЄ N b) a* b = a - b , a,bЄ Z c) a* b = a /b , a,bЄ N d) a * b = HCF of a and b , a,bЄ N e) a * b =(a + b) / (a- b) on Z f) a * b = (a x b) / 5 on Q g) a * b = ( a + b ) 2 on Z 2. Give reasons for each of the following statements. a) Subtraction is not a binary operation on R+ b) If a * b = a,bЄ R+ then * is a binary operation. c) Division is a binary operation on non zero rational numbers d) Division is not a binary operation on Q 3. Which of the following is an Algebraic Structure ? a) Set of Natural numbers with respect to Division operation. b) Set of Real numbers w.r.t Division operation. c) Set of odd integers w.r.t Division operation. d) Set of even integers w.r.t Multiplication operation. e) Set of even integers w.r.t operation * defined by a * b = , for all a,bЄ Z Activity : Select any set and an operation . Check whether the set is an algebraic structure under that operation. Do this for many sets.

  15. END OF CHAPTER

More Related