{"id":29224,"date":"2022-03-28T17:00:40","date_gmt":"2022-03-28T11:30:40","guid":{"rendered":"https:\/\/mcq-questions.com\/?p=29224"},"modified":"2022-03-28T17:22:10","modified_gmt":"2022-03-28T11:52:10","slug":"ncert-solutions-for-class-12-maths-chapter-1-ex-1-4","status":"publish","type":"post","link":"https:\/\/mcq-questions.com\/ncert-solutions-for-class-12-maths-chapter-1-ex-1-4\/","title":{"rendered":"NCERT Solutions for Class 12 Maths Chapter 1 Relations and Functions Ex 1.4"},"content":{"rendered":"

These NCERT Solutions for Class 12 Maths<\/a> 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\/<\/p>\n

NCERT Solutions for Class 12 Maths Chapter 1 Relations and Functions 1.4<\/h2>\n

\"NCERT<\/p>\n

Ex 1.4 Class 12\u00a0 Question 1.<\/strong>
\nDetermine whether or not each of the definition of * given below gives a binary operation. In the event that c is not a binary operation, given justification for this.
\ni. On Z+<\/sup> define * by a * b = a – b
\nii. On Z+<\/sup>, define * by a * b = ab
\niii. On R, define * by a * b = ab\u00b2
\niv. On Z+<\/sup>, define * by a * b = |a – b|
\nv. On Z+<\/sup>, define * by a * b = a
\nSolution:
\ni. Let 3, 4 \u2208 Z+<\/sup>
\n3 * 4 = 3 – 4 = – 1 \u2208 Z+<\/sup>
\nHence * is not a binary operation.<\/p>\n

ii. For every element a, b \u2208 Z+<\/sup>, ab \u2208 Z+<\/sup>
\nHence * is a binary operation.
\nExample : 3, 4 \u2208 Z+<\/sup>, 3 *4 = 3 x 4= 12 \u2208 Z+<\/sup><\/p>\n

iii. For a, b \u2208 R, ab\u00b2 \u2208 R and is unique.
\nHence a * b = ab\u00b2 is a binary operation,
\nExample : 5, b \u2208 R, 5 * 6 = 5 x 6\u00b2 = 180 \u2208 R<\/p>\n

iv. For a, b \u2208 Z+<\/sup>, |a – b| \u2208 Z+<\/sup>
\nHence * is a binary operation.
\nExample : 3, 4 \u2208 Z+<\/sup>, 3 * 4 = |3 – 4| = |- 1| = 1 \u2208 Z+<\/sup><\/p>\n

v. For a, b \u2208 Z+<\/sup>+, a * b = 2 \u2208 Z+<\/sup>
\nHence * is a binary operation.
\nExample : 2, 3 \u2208 Z+<\/sup>, 2 * 3 = 2 \u2208 Z+<\/sup><\/p>\n

Class 12 Maths Chapter 1 Exercise 1.4 Question 2.<\/strong>
\nFor each binary operation * defined below, determine whether * is commutative or associative.
\ni. On Z+<\/sup>, define a * b = a – b
\nii. On Q, define a * b = ab + 1
\niii. On Q, define a * b = \\(\\frac { ab }{ 2 }\\)
\niv. On Z+<\/sup>, define a * b = 2ab<\/sup>
\nv. On Z+<\/sup>, define a * b = ab<\/sup>
\nvi. On R – {- 1}, define a * b = \\(\\frac { a }{ b+1 }\\)
\nSolution:
\ni. a * b = a – b; b * a = b – a
\n\u2234 a * b * b * a
\n\u2234 * is not commutative<\/p>\n

Example:
\n2 * 3 = 2 – 3 = – 1 and 3 * 2 = 3 – 2 = 1
\na * (b * c) = a – (b – c) = a – b + c
\n(a * b) * c = (a – b) – c = a – b – c
\n\u2234 a * (b * c) \u2260 (a * b) * c
\n\u2234 * is not associative<\/p>\n

ii. a * b = ab + 1
\nb * a = ba + 1 = ab + 1 since ab – ba in Q.
\n\u2234 a * b = b * a.
\nHence * is commutative.
\na * (b * c) = a * (bc + 1)
\n= a(bc + 1) + 1 = abc + a + 1
\n(a * b) * c = (ab + 1) * c
\n= (ab + 1)c + 1 = abc + c + 1
\n\u2234 a * (b * c) \u2260 (a * b) * c
\n* is not associative.<\/p>\n

\"NCERT<\/p>\n

iv. a * b = 2ab<\/sup>
\nb * a = 2ba<\/sup> = 2ab<\/sup> since ab = ba in Z+<\/sup>
\n\u2234 a * b = b * a
\n\u2234 * is commutative
\na * (b * c) = a * 2bc<\/sup> = 2a(2bc<\/sup>)<\/sup>
\n(a * b) * c = 2 ab * c = 2(2ab)<\/sup>c
\n\u2234 a * (b * c) \u2260 (a * b) * c.
\n\u2234 * is not associative<\/p>\n

v. a * b = ab<\/sup>
\nb * a = bb<\/sup> Since ab<\/sup> \u2260 ba<\/sup>, a * b \u2260 b * a
\n\u2234 * is not commutative.<\/p>\n

Example :
\n2 * 3 = 23<\/sup>3 = 8
\n3 * 2 = 3\u00b2 = 9
\n\u2234 2 * 3 \u2260 3 * 2
\na * (b * c) = a * (bc<\/sup>) = a(bc<\/sup><\/sup> )
\n(a * b) * c = (ab) * c = (ab<\/sup>)c<\/sup> = abc<\/sup>
\n\u2234 a * (b * c) \u2260 (a * b) * c
\n\u2234 * is not associative<\/p>\n

\"NCERT
\nHence * is not associative.<\/p>\n

\"NCERT<\/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\"NCERT<\/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\"NCERT
\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\"NCERT
\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

\"NCERT<\/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\"NCERT<\/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

\"NCERT<\/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>\n

Question 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

\"NCERT<\/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

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