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# hw22 - xy The ﬁrst two rows have been completed for you...

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Intro to Abstract Math Homework 22 Fall 2009 Due November 4 Exercise 63. Prove or disprove the following statements. a. Subtraction is a binary operation on Z . b. Subtraction is a binary operation on N . c. Division is a binary operation on N . d. Division is a binary operation on Q . Exercise 64. Let Id = 1 2 3 1 2 3 , α = 1 2 3 1 3 2 , β = 1 2 3 3 2 1 , γ = 1 2 3 2 1 3 , δ = 1 2 3 2 3 1 , = 1 2 3 3 1 2 denote the elements of S 3 . We have seen that function composition is a binary operation on S 3 . Complete the following “composition table,” which gives the results of composing any two elements of S 3 . The entry in the x row and y column is xy . The first two rows have
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Unformatted text preview: xy . The ﬁrst two rows have been completed for you as an example. Id α β γ δ ± Id Id α β γ δ ± α α Id δ ± β γ β γ δ ± Exercise 65. If we let R θ denote counterclockwise rotation by θ degrees, H denote the ﬂip across the vertical axis, V denote the ﬂip across the horizontal axis, and F i ( i = 1 , 2) denote the diagonal ﬂips, then recall that the complete set if symmetries of the square is D 4 = { R ,R 90 ,R 180 ,R 270 ,V,H,F 1 ,F 2 } . Compute the “composition table” for D 4 as you did for S 3 above....
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