{"id":29598,"date":"2024-12-06T05:25:47","date_gmt":"2024-12-05T23:55:47","guid":{"rendered":"https:\/\/samacheerkalvi.guide\/?p=29598"},"modified":"2024-12-07T10:15:16","modified_gmt":"2024-12-07T04:45:16","slug":"samacheer-kalvi-12th-maths-guide-chapter-12-ex-12-1","status":"publish","type":"post","link":"https:\/\/samacheerkalvi.guide\/samacheer-kalvi-12th-maths-guide-chapter-12-ex-12-1\/","title":{"rendered":"Samacheer Kalvi 12th Maths Guide Chapter 12 Discrete Mathematics Ex 12.1"},"content":{"rendered":"

Tamilnadu State Board New Syllabus\u00a0Samacheer Kalvi 12th Maths Guide<\/a> Pdf Chapter 12 Discrete Mathematics Ex 12.1 Textbook Questions and Answers, Notes.<\/p>\n

Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.1<\/h2>\n

Question 1.
\nDetermine whether * is a binary operation on the sets-given below
\n(i) a * b – a. |b| on R
\n(ii) a * b = min (a, b) on A = {1, 2, 3, 4, 5}
\n(iii) (a * b) = a\u221ab is binary on R
\nSolution:
\n(i) Yes.
\nReason: a, b \u2208 R. So, |b| \u2208 R when b \u2208 R
\nNow multiplication is binary on R
\nSo a|b| \u2208 R when a, b \u2208 R.
\n(Le.) a * b \u2208 R.
\n* is a binary operation on R.<\/p>\n

(ii) Yes.
\nReason: a, b \u2208 R and minimum of (a, b) is either a or b but a, b \u2208 R.
\nSo, min (a, b) \u2208 R.
\n(Le.) a * b \u2208 R.
\n* is a binary operation on R.<\/p>\n

(iii) a* b = \\(a \\sqrt{b}\\) where a, b \u2208 R.
\nNo. * is not a binary operation on R.
\nReason: a, b \u2208 R.
\n\u21d2 b can be -ve number also and the square root of a negative number is not real.
\nSo \\(\\sqrt{b}\\) \u2209 R even when b \u2208 R.
\nSo \\(\\sqrt{b}\\) \u2209 R. ie., a * b \u2209 R.
\n* is not a binary operation on R.<\/p>\n

\"Samacheer<\/p>\n

Question 2.
\nOn Z, define \u2297 by (m \u2297 n) =m\u207f + nm<\/sup>: \u2200m, n \u2208 Z Is \u2297 binary on Z?
\nSolution:
\nNo. * is not a binary operation on Z.
\nReason: Since m, n \u2208 Z.
\nSo, m, n can be negative also.
\nNow, if n is negative (Le.) say n = -k where k is +ve.
\n\"Samacheer
\nSimilarly, when m is negative then nm<\/sup> \u2209 Z.
\n\u2234 m * n \u2209 Z. \u21d2 * is not a binary operation on Z.<\/p>\n

Question 3.
\nLet * be defined on R by (a * b) = a + b + ab – 7. Is * binary on R? If so, find 3 * (\\(\\frac { -7 }{ 15}\\))
\nSolution:
\n(a * b) = a + b + ab – 7 \u2200 a, b \u2208 R
\nIf a \u2208 R, b \u2208 R then ab \u2208 R
\n\u2234 (a * b) = a + b + ab – 7 \u2208 R
\nFor example, let 1, 2 \u2208 R
\n(1 * 2) = 1 + 2 + (1)(2) – 7
\n= -2 \u2208 R
\n\u2234 * is a binary operation on R
\n\"Samacheer<\/p>\n

\"Samacheer<\/p>\n

Question 4.
\nLet A = {a + \u221a5 b: a, b \u2208 Z}. Check whether the usual multiplication is a binary operation on A.
\nSolution:
\nLet A = a + \\(\\sqrt{5}\\) b and B = c + \\(\\sqrt{5}\\)d, where a, b, c, d \u2208 M.
\nNow A * B ={a + \\(\\sqrt{5}\\)b)(c + \\(\\sqrt{5}\\)d)
\n= ac + \\(\\sqrt{5}\\)ad + \\(\\sqrt{5}\\)bc + \\(\\sqrt{5}\\)b\\(\\sqrt{5}\\)d
\n= (ac + 5bd) + \\(\\sqrt{5}\\)(ad+ bc) \u2208 A
\nWhere a, b, c, d \u2208 Z
\nSo * is a binary operation.<\/p>\n

Question 5.
\n(i) Define an operation * on Q as follows:
\na * b = (\\(\\frac { a+b }{ 2}\\)); a, b \u2208 Q. Examine the closure, commutative and associate properties satisfied by * on Q.
\n(ii) Define an operation * on Q as follows: a * b = (\\(\\frac { a+b }{ 2}\\)); a, b \u2208 Q. Examine the existence of identity and the existence of inverse for the operation * on Q.
\nSolution:
\n\"Samacheer
\nso, a * (b * c) \u2260 (a * b) * c
\nHence, the binary operation * is not associative.<\/p>\n

\"Samacheer<\/p>\n

(ii) a * b = (\\(\\frac { a+b }{ 2}\\)); a, b \u2208 Q
\nFor identity, a * e = e * a = a
\nNow; a * e = a
\n\\(\\frac { a+e }{ 2}\\) = a
\na + e = 2a
\ne = 2a – a = a
\nWhich is not possible
\n\u2234 Identity does not exist and hence the inverse does not exist.<\/p>\n

Question 6.
\nFill in the following table so that the binary operation * on A = {a, b, c} is commutative.
\n\"Samacheer
\nSolution:
\nGiven * is commutative on A = {a, b, c}
\n\"Samacheer
\nFrom the table, it is given that b * a = c
\n\u21d2 a * b = c
\nas * is commutative
\nc * a = a = a * c and
\nb * c = a = c * b<\/p>\n

\"Samacheer<\/p>\n

Question 7.
\nConsider the binary operation * defined on the set A = {a, b, c, d} by the following table:
\n\"Samacheer
\nSolution:
\n* is defined on the set A = {a, b, c, d} Given table
\n\"Samacheer
\nFrom the table a * b = c and b * a = d
\n\u21d2 a * b \u2260 b * a
\n\u2234 * is not commutative.
\nNow(a * b) * c = c * c = a
\na * (b * c) = a * b = c
\n\u21d2 (a * b) * c \u2260 a * (b * c)
\n\u2234 * is not associative<\/p>\n

Question 8.
\nLet
\n\"Samacheer
\nbe any three boolean matrices of the same type. Find
\n(i) A v B
\n(ii) A \u2227 B
\n(iii) (A v B) \u2227 C
\n(iv) (A \u2227 B) v C.
\nSolution:
\nGiven boolean matrices
\n\"Samacheer
\n\"Samacheer<\/p>\n

\"Samacheer<\/p>\n

Question 9.
\n(i) Let M = \\(\\left\\{\\left(\\begin{array}{ll}
\nx & x \\\\
\nx & x
\n\\end{array}\\right): x \\in R-\\{0\\}\\right\\}\\)\u00a0and let * be the matrix multiplication. Determine whether M is closed under *. If so, examine the commutative and associative properties satisfied by * on M.
\n(ii) Let M = \\(\\left\\{\\left(\\begin{array}{ll}
\nx & x \\\\
\nx & x
\n\\end{array}\\right): x \\in R-\\{0\\}\\right\\}\\)\u00a0and let * be the matrix multiplication. Determine whether M is closed under *. If so, examine the existence of identity, existence of inverse properties for the operation * on M
\nSolution:
\n\"Samacheer
\nSince x \u2260 0, y \u2260 0, we see that 2xy \u2260 0 and so AB \u2208 M. This shows that M is closed under matrix multiplication.
\nCommutative axiom:
\nAB = \\(\\left(\\begin{array}{ll}
\n2xy & 2xy \\\\
\n2xy & 2xy
\n\\end{array}\\right)\\)
\n= BA for all A, B \u2208 M
\nHere, Matrix multiplication is commutative (though in general, matrix multiplication is not commutative)
\nAssociative axiom:
\nSince matrix multiplication is associative, this axiom holds goods for M.<\/p>\n

(ii) Identity axiom:
\n\"Samacheer
\nAlso, we can show that EA = A
\nHence E is the identity element in M
\n\"Samacheer<\/p>\n

\"Samacheer<\/p>\n

Question 10.
\n(i) Let A be Q\\{1} Define * on A by x * y = x + y – xy. Is * binary on A ? If so, examine the commutative and associative properties satisfied by * on A.
\n(ii) Let A be Q\\{1}. Define * on A by x * y = x + y – xy. Is * binary on A ? If so, examine the existence of an identity, the existence of inverse properties for the operation * on A.
\nSolution:
\n(i) Let a, b \u2208 A (i.e.) a \u2260 \u00b11 , b \u2260 1
\nNow a * b = a + b – ab
\nIf a + b – ab = 1 \u21d2 a + b – ab – 1 = 0
\n(i.e.) a(1 – b) – 1(1 – b) = 0
\n(a – 1)(1 – b) = 0 \u21d2 a = 1, b = 1
\nBut a \u2260 1 , b \u2260 1
\nSo (a – 1) (1 – 6) \u2260 1
\n(i.e.) a * b \u2208 A. So * is a binary on A.<\/p>\n

To verify the commutative property:<\/p>\n

Let a, b \u2208 A (i.e.) a \u2260 1 , b \u2260 1
\nNow a * b = a + b – ab
\nand b * a = b + a – ba
\nSo a * b = b * a \u21d2 * is commutative on A.<\/p>\n

To verify the associative property:
\nLet a, b, c \u2208 A (i.e.) a, b, c \u2260 1
\nTo prove the associative property we have to prove that
\na * (b * c) = (a * b) * c<\/p>\n

LHS: b * c = b + c – bc = D(say)
\nSo a * (b * c) = a * D = a + D – aD
\n= a + (b + c – bc) – a(b + c – bc)
\n= a + b + c – bc – ab – ac + abc
\n= a + b + c – ab – bc – ac + abc …… (1)<\/p>\n

RHS: (a * b) = a + b – ab = K(say)
\nSo (a * b) * c = K * c = K + c – Kc
\n= (a + b – ab) + c – (a + b – ab) c
\n= a + b – ab + c – ac – bc + abc
\n= a + b + c – ab – bc – ac + abc ….. (2)<\/p>\n

(ii) To verify the identity property:
\nLet a \u2208 A (a \u2260 1)
\nIf possible let e \u2208 A such that
\na * e = e * a = a
\nTo find e:
\na * e = a
\n(i.e.) a + e – ae = a
\n\"Samacheer
\nSo, e = (\u2260 1) \u2208 A
\n(i.e.) Identity property is verified.
\nTo verify the inverse property:
\nLet a \u2208 A (i.e. a \u2260 1)
\nIf possible let a’ \u2208 A such that
\nTo find a’:
\na * a’ = e
\n(i.e.) a + a’ – aa’ = 0
\n\u21d2 a'(1 – a) = – a
\n\"Samacheer
\n\u21d2 For every \u2208 A there is an inverse a’ \u2208 A such that
\na* a’ = a’ * a = e
\n\u21d2 Inverse property is verified.<\/p>\n

\"Samacheer<\/p>\n","protected":false},"excerpt":{"rendered":"

Tamilnadu State Board New Syllabus\u00a0Samacheer Kalvi 12th Maths Guide Pdf Chapter 12 Discrete Mathematics Ex 12.1 Textbook Questions and Answers, Notes. Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.1 Question 1. Determine whether * is a binary operation on the sets-given below (i) a * b – a. |b| on R …<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","enabled":false},"version":2}},"categories":[5],"tags":[],"class_list":["post-29598","post","type-post","status-publish","format-standard","hentry","category-class-12"],"jetpack_publicize_connections":[],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/samacheerkalvi.guide\/wp-json\/wp\/v2\/posts\/29598"}],"collection":[{"href":"https:\/\/samacheerkalvi.guide\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/samacheerkalvi.guide\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/samacheerkalvi.guide\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/samacheerkalvi.guide\/wp-json\/wp\/v2\/comments?post=29598"}],"version-history":[{"count":1,"href":"https:\/\/samacheerkalvi.guide\/wp-json\/wp\/v2\/posts\/29598\/revisions"}],"predecessor-version":[{"id":41475,"href":"https:\/\/samacheerkalvi.guide\/wp-json\/wp\/v2\/posts\/29598\/revisions\/41475"}],"wp:attachment":[{"href":"https:\/\/samacheerkalvi.guide\/wp-json\/wp\/v2\/media?parent=29598"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/samacheerkalvi.guide\/wp-json\/wp\/v2\/categories?post=29598"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/samacheerkalvi.guide\/wp-json\/wp\/v2\/tags?post=29598"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}