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This contradicts the assumption that p1 , p2 ,. By induction, S has 2n subsets that do not contain a. But there is one-to-one correspondence between the subsets of S that do not contain a and those that do.
Use induction on n. Mimic Example We must prove that S contains all integers greater than or equal to a. Let T be the set of all integers greater than a that are not in S and suppose that T is not empty. This contradicts our assumption that b is not in S. Observe that the number with the decimal representation a9 a8. No For the case in which the check digit is not involved, the argument given Exercise 41 applies to transposition errors.
Denote the money order number by a9 a8. In this case the transposition does not yield an error. Say the number is a8 a7. Say that the weight for a is i. Say the valid number is a1 a2. Since the only pair is 39, the correct number is First note that the sum of the digits modulo 11 is 2. So, some digit is 2 too large. Say the error is in position i. Thus, the digit in position 5 to 2 too large.
So, the correct number is An error in an even numbered position changes the value of the sum by an even amount. The equivalence classes can be represented by the real numbers in the interval [0, 1. The equivalence classes are the set of even integers and the set of odd integers. So, a and c belong to A. The last digit of is the value of mod Similarly, the last digit of is the value of mod R c. R , H, V, D, D0 e. In addition, Dn has n reflections.
When n is odd, the axes of reflection are the lines from the vertices to the midpoints of the opposite sides. When n is even, half of the axes of reflection are obtained by joining opposite vertices; the other half, by joining midpoints of opposite sides. A nonidentity rotation leaves only one point fixed — the center of rotation.
A reflection leaves the axis of reflection fixed. A reflection followed by a different reflection would leave only one point fixed the intersection of the two axes of reflection so it must be a rotation. A rotation followed by a rotation either fixes every point and so is the identity or fixes only the center of rotation. However, a reflection fixes a line. In either case, the set of points fixed is some axis of reflection.
Dn is not commutative. Let the distance from a point on one H to the corresponding point on an adjacent H be one unit. All other symmetries are compositions of finitely many of those already described. The group is non-Abelian. In each case the group is D6. D28 D22 Their only symmetry is the identity. It is symmetric under a horizontal reflection. The set does not contain the identity; closure fails. Under multiplication modulo 4, 2 does not have an inverse. Modulo multiplication is associative.
Use D4. Let g belong to G. The identity is First note that if 1 is in H then by closure all five elements are in H. Matrix 1 0 multiplication is associative. In a non-Abelian group ab n need not equal an bn. Use D4 for the examples.
Suppose that G is Abelian. Now multiply on the left by a. Z ; Z44 and D Suppose x appears in a row labeled with a twice. But we use distinct elements to label the columns. Proceed as follows. By definition of the identity, we may complete the first row and column.
Then complete row 3 and column 5 by using Exercise In row 2 only c and d remain to be used. This observation allows us to complete row 2. Then rows 3 and 4 may be completed by inserting the unused two elements. Finally, we complete the bottom row by inserting the unused column elements.
Use cancellation. Since e is one solution it suffices to show that nonidentity solutions come in distinct pairs. So if we can find one nonidentity solution we can find a second one. Thus, finding a third nonidentity solution gives a fourth one. R5 F Now cancel on left and right. Now, using cancellation we have that a2 , a3 , a4 are not the identity and are distinct from each other and distinct from a. If b is another solution that is not a power of a, then by the same argument b, b2 , b3 and b4 are four distinct nonidentity solutions.
We must further show that b2 , b3 and b4 are distinct from a, a2 , a3 , a4. A similar argument applies to b3 and b4. In the general case, the number of solutions is a multiple of 4 or is infinite. This c d happens when a and d are 1 and at least 1 of b and c is 0 and when b and c are 1 and at least 1 of a and d is 0. In each case, notice that the order of the element divides the order of the group.
The infinite case follows from the finite case. So, 2 and 28 are inverses of each other and 8 and 22 are inverses of each other.
So, 2 and 8 are inverses of each other and 7 and 13 are inverses of each other.
Joseph Gallian Solutions manual to Contemporary Abstract Algebra. 2012
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Contemporary Abstract Algebra Solutions Manual Pdf