ANSWERS A-29 68. a) 68. a) It is closed; identity element is F; inverses: − A E, − B B, − − − − C C D D E A F F , , , ; is associative; for example, C D A C D A E A C C F F ( ) ( ) ? ? = = = b) B C C B E A ?= ≠ 69. a) * R S T U V I R V T U S I R S U I V R T S T S R I V U T U T V R I S U V I U S T R V I R S T U V I b) Yes. It is closed; identity element is I; inverses R V, − SSTTUUV RII , , , , ; − − − − − and the associative property will hold. SECTION 9.3, PAGE 566 65. a) + 0 1 2 0 0 1 2 1 1 2 0 2 2 0 1 66. a) + 0 1 2 3 0 0 1 2 3 1 1 2 3 0 2 2 3 0 1 3 3 0 1 2 b) Yes c) Yes, 0 d) Yes; − − − − 0 0,1 3,2 2,3 1 e) (12)31(23) 3 3 1 1 2 2 ? ? + + = + + + = + = f) Yes; + = + = 2 1 1 2 3 3 g) Yes. All five properties are satisfied. h) Yes 67. a) × 0 1 2 3 0 0 0 0 0 1 0 1 2 3 2 0 2 0 2 3 0 3 2 1 × 0 1 2 3 4 5 6 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 2 0 2 4 6 1 3 5 3 0 3 6 2 5 1 4 4 0 4 1 5 2 6 3 5 0 5 3 1 6 4 2 6 0 6 5 4 3 2 1 b) Yes c) Yes; 1 d) No; the element 0 does not have an inverse. e) (12)41(24) 2 4 1 1 1 1 ? ? × × = × × × = × = f) Yes; × = × 2 3 3 2 = 6 6 g) No; 0 does not have an inverse. SECTION 9.4, PAGE 577 12. − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ 1 9 4 6 8 5 5 3 13 27. + = − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ × = − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ A B A B 1 5 6 7 ; 32 15 24 12 28. + = ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ × = − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ A B A B 1 6 9 3 ; 7 18 27 10 29. + = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ A B 8 1 3 4 3 0 ; cannot be multiplied 30. + = ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ A B 12 10 8 6 4 2 ; cannot be multiplied 31. Cannot be added; × = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ A B 24 34 24 34 36. + = + = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ A B B A 11 4 0 13 37. + = + = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ A B B A 8 0 6 8 38. + = + = − − − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ A B B A 7 0 2 8 1 5 2 1 17 39. + + = + + = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ A B C A B C ( ) ( ) 7 13 2 10 40. + + = + + = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ A B C A B C ( ) ( ) 5 5 7 37
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