bc<\/sup> \n\u2234 a * (b * c) \u2260 (a * b) * c \n\u2234 * is not associative<\/p>\n \nHence * is not associative.<\/p>\n
<\/p>\n
Question 3. \nConsider the binary operation ^ on the set {1,2,3,4, 5} defined by a ^ b = min {a, b}. Write the operation table of the operation ^ . \nSolution: \n1 ^ 1 = min{1, 1} = 1, 1 ^ 2 = min{1, 2} = 1, etc. \n2 ^ 1 = min{2, 1} = 1, 2 ^ 2 = min{2, 2} = 2, etc. \n3 ^ 1 = min{3,1} = 1, 3 ^ 2 = min{3, 2} = 2, etc. and so on. \nThe operation table of the operation. \n <\/p>\n
Question 4. \nConsider a binary operation * on the set {1, 2, 3, 4, 5} given by the following multiplication table \ni. Compute (2 * 3) * 4 and 2 * (3 * 4) \nii. Is * commutative? \niii. Compute (2 * 3) * (4 * 5). \n \nSolution: \ni. (2 * 3) * 4 = 1 * 4 = 1 \n2 * (3 * 4) = 2 * 1 = 1<\/p>\n
ii. The entries in the table are symmetric along the main diagonal. Hence * is commutative.<\/p>\n
iii. (2 * 3) * (4 * 5) = 1 * 1 = 1<\/p>\n
Question 5. \nLet *’ be the binary operation on the set {1, 2, 3, 4, 5} defined by a *’ b = H.C.F. of a and b. Is the operation *’ same as the operation * defined in Question 4 above? Justify your answer. \nSolution: \n \n*’ gives the same table as given in. Hence * and *’ are the same operations.<\/p>\n
Question 6. \nLet * be the binary operation on N given by a * b = L.C.M. of a and b. Find \ni. 5 * 7, 20 * 16 \nii. Is * commutative? \niii. Is * associative? \niv. Find the identity of * in N \nv. Which elements of N are invertible for the operation *? \nSolution: \ni. 5 * 7 = LCM (5, 7) = 5 x 7 = 35 \n20 * 16 = LCM (20, 16) = 80<\/p>\n
ii. a * b = LCM (a, b) \nb * a = LCM (b, a) = LCM (a, b) \n\u2234 a * b = b * a \n\u2234 * is commutative<\/p>\n
iii. a * (b * c) = a * LCM (b, c) \n= LCM (a, b, c) \n(a * b) * c = LCM (a, b) * c \n= LCM (a, b, c) \n\u2234 a * (b * c) = (a * b) * c \n\u2234 * is associative<\/p>\n
iv. Let e be the identity element in N. \nThen a * e = e * a = a, for all a \u2208 N \n\u21d2 LCM (a, e) = a, for all a \u2208 N \n\u21d2 e = 1 \u2208 N \n\u2234 1 is the identity element of * in N.<\/p>\n
v. Let a be an invertible element in N. \nThen there exists an element b in N such that \na * b = 1 = b * a \n\u21d2 LCM (a, b)= 1 \na = b= 1 \nThus 1 is an invertible element in N.<\/p>\n
Question 7. \nIs * defined on the set {1, 2, 3, 4, 5} by a * b = L.C.M. of a and b a binary operation? Justify your answer. (March 2016) \nSolution: \na * b = LCM (a, b) \nNow 2 * 3 = LCM (2, 3) = 6 \nBut 6 is not an element of the given set. \nHence * is not a binary operation.<\/p>\n
<\/p>\n
Question 8. \nLet * be the binary operation on N defined by a * b = H.C.F. of a and b. \nIs * commutative? Is * associative? \nDoes there exist identity for this binary operation on N? (March 2013) \nSolution: \na * b = HCF (a, b) for all a, b \u2208 N \nb * a = HCF (b, a) for all a, b \u2208 N \n\u2234 a * b = b * a \n\u2234 * is commutative \na * (b * c) = a * HCF (b, c) \n= HCF (a, b, c) \n(a * b) * c = HCF (a, b) * c = HCF (a, b, c) \nHence a * (b * c) = (a * b) * c \n\u2234 * is associative \nLet e be the identity element in N. \n\u21d2 a * e = e * a = a for all a \u2208 N \n\u21d2 HCF (a, e) = HCF (e, a) = a for all a \u2208 N \n\u21d2 There is no e e N which makes this true. \n\u21d2 Identity element does not exist in N.<\/p>\n
Question 9. \nLet * be a binary operation on the set Q of rational numbers as follows: \ni. a * b = a – b \nii. a * b = a\u00b2 + b\u00b2 \niii. a * b = a + ab \niv. a * b = (a – b)\u00b2 \nv. a * b = \\(\\frac { ab }{ 4 }\\) \nvi. a * b = ab\u00b2 \nFind which of the binary operations are commutative and which are associative. \nSolution: \ni. a * b = a – b b * a = b – a \nHence a * b \u2260 b * a \n\u2234 * is not commutative \na * (b * c) = a* (b – c) \n= a – (b – c) = a – b + c \n(a * b) * c = (a – b) * c = (a – b) – c = a – b – c \nHence (a * b) * c \u2260 a * (b * c) \n\u2234 * is not associative.<\/p>\n
ii. a * b = a\u00b2 + b\u00b2 \nb * a = b\u00b2 + a\u00b2 = a\u00b2 + b\u00b2 = a * b \n\u2234 * is commutative. \na * (b * c) = a * (b\u00b2 + c\u00b2) = a\u00b2 + (b\u00b2 + c\u00b2)\u00b2 \n(a * b) * c = (a\u00b2 + b\u00b2) * c \n= (a\u00b2 + b\u00b2)\u00b2 + c\u00b2 \u2260 a\u00b2 + (b\u00b2 + c\u00b2)\u00b2 \n\u2234 * is not associative<\/p>\n
iii. a * b = a + ab; \nb * a = b + ba \na * b \u2260 b * a \n\u2234 * is not commutative. \na * (b * c) = a * (b + bc) \n= a + a(b + bc) = a + ab + abc \n(a * b) * c = (a + ab) * c \n= (a + ab) + (a + ab)c \n= a + ab + ac + abc \na*(b + c) \u2260 (a * b) * c \n\u2234 * is not associative.<\/p>\n
iv. a * b = (a – b)\u00b2 = a\u00b2 – 2ab + b\u00b2 \nb * a = (b – a)\u00b2 = b\u00b2 – 2ab + a\u00b2 \n= a\u00b2 – 2ab + b\u00b2 = a * b \n\u2234 * is commutative. \na * (b * c) = a * (b – c)\u00b2 = [a – (b – c)\u00b2]\u00b2 \n(a * b) * c = (a – b)\u00b2 * c = [(a – b)\u00b2 – c]\u00b2 \na * (b * c) \u2260 (a * b) * c \n\u2234 * is not associative.<\/p>\n
v. a * b = \\(\\frac { ab }{ 4 }\\) \nb * a = \\(\\frac { ba }{ 4 }\\) = \\(\\frac { ab }{ 4 }\\) = a * b \n\u2234 * is commutative. \n <\/p>\n
vi. a * b = ab\u00b2 ; b * a = ba\u00b2 \na * b \u00b1 b * a \n\u2234 * is not commutative. \na * (b * c) = a * (bc\u00b2) = a(bc\u00b2)\u00b2 = ab\u00b2c4<\/sup> \n(a * b) * c = (ab\u00b2) * c = (ab\u00b2)(c\u00b2) = ab\u00b2c\u00b2 \na * (b * c) \u2260 (a * b) * c \n\u2234 * is not associative.<\/p>\n <\/p>\n
Question 10. \nShow that none of the operations given in Question No. 9 (except (v)) has identity. \nSolution: \ni. Let e be the identity element. \na * e = e * a = a \n\u21d2 a\u00b2 + e\u00b2 = e\u00b2 + a\u00b2 = a \nwhich is not possible for any e \u2208 Q. \n\u2234 There is no identity element.<\/p>\n
ii. Let e be the identity element. \na * e = e * a = a \n\u21d2 (a – e)\u00b2 = (e – a)\u00b2 = a \nwhich is not possible for any e \u2208 Q. \n\u2234 There is no identity element. .<\/p>\n
iii. Let e be the identity element. \na * e = e * a = a \n\u21d2 ae\u00b2 = ea\u00b2 = a \nwhich is not possible for any e \u2208 Q. (SAY 2014) \n\u2234 There is no identity element.<\/p>\n
Question 11. \nLet A = N x N and * be the binary operation on A defined by (a, b) * (c, d) = (a + c, b + d) \nShow that * is commutative and associative. Find the identity element for * on A, if any. \nSolution: \n(a, b) * (c, d) = (a + c, b + d) \n(c, d) * (a, b) = (c + a, d+ b) = (a + c, b + d) \n(a, b) * (c, d) = (c, d) * (a, b) \n\u2234 * is commutative. \nLet (a, b), (c, d), (e, f) \u2208 A. \n(a, b) * [(c, d) * (e, f)] \n= (a, b) * [(c + e, d +f)] \n= (a + c + e, b + d +f) \n[(a, b) * (c, d)] * (e, f) = (a + c, b + d) * (e, f) \n= (a + c + e, b + d + f) \ni.e., (a, b) * [(c, d) * (e, f)] = [(a, b) * (c, d)] * (e, f) \n\u2234 * is associative. \nHere identity element does not exist. \nLet (e1<\/sub>, e2<\/sub>) \u2208 be the identity element for * in A. \n\u2234 (a, b) * (e1<\/sub> + e2<\/sub>) = (e1<\/sub>, e2<\/sub>) * (a, b) = (a, b) \n\u21d2 (a + e1<\/sub>, b + e2<\/sub>) = (e1<\/sub> + a, e2<\/sub> + b) = (a, b) \n\u21d2 a + e1<\/sub> = a and b + e2<\/sub> = b \n\u21d2 e1<\/sub> = 0 and e2<\/sub> = 0 \n(e1<\/sub>, e2<\/sub>) = (0, 0) \u2209 A, since A = N x N \nHence there is no identity element for * in A.<\/p>\nQuestion 12. \nState whether the following statements are true or false. Justify. \ni. For an arbitrary binary operation * on a set N, a * a = a for all a \u2208 N \nii. If * is a commutative binary operation on N, then \na * (b * c) = (c * b) * a. \nSolution: \ni. False \nLet a * b = a + b, a, b \u2208 N \n\u2234 a * a = a + a = 2a \u2260 a<\/p>\n
ii. True \nSince * is commutative b * c = c * b \n\u2234 a * (b * c) = a * (c * b) = (c * b) * a<\/p>\n
<\/p>\n
Question 13. \nConsider a binary operation * on N defined as a * b = a\u00b3 + b\u00b3. Choose the correct answer. \na. Is * both associative and commutative? \nb. Is * commutative but not associative? \nc. Is * associative but not commutative? \nd. Is * neither commutative nor associative? \nSolution: \nb. Is * commutative but not associative? \na * b = a\u00b3 + b\u00b3 = b\u00b3 + a\u00b3 = b * a \n\u2234 * is commutative. \na * (b * c ) = a * (b\u00b3 + c\u00b3) = a\u00b3 + (b\u00b3 + c\u00b3)\u00b3 \n(a * b) * c = (a\u00b3 + b\u00b3) * c = (a\u00b3 + b\u00b3)\u00b3 + c\u00b3 \n* is not associative.<\/p>\n","protected":false},"excerpt":{"rendered":"
These NCERT Solutions for Class 12 Maths Chapter 1 Relations and Functions Ex 1.4 Questions and Answers are prepared by our highly skilled subject experts. https:\/\/mcq-questions.com\/ncert-solutions-for-class-12-maths-chapter-1-ex-1-4\/ NCERT Solutions for Class 12 Maths Chapter 1 Relations and Functions 1.4 Ex 1.4 Class 12\u00a0 Question 1. Determine whether or not each of the definition of * given …<\/p>\n
NCERT Solutions for Class 12 Maths Chapter 1 Relations and Functions Ex 1.4<\/span> Read More »<\/a><\/p>\n","protected":false},"author":9,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"default","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","spay_email":""},"categories":[3],"tags":[],"yoast_head":"\nNCERT Solutions for Class 12 Maths Chapter 1 Relations and Functions Ex 1.4 - MCQ Questions<\/title>\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\t \n\t \n\t \n