Question 1 Which of the following is a nontrivial subgroup of where is the additive group of integers modulo 3 consisting of { 0,1,2 }, and the Cartesian product is the set of ordered pairs? A) { (0,0), (1,1), (2,2) } B) { (0,0) , (1,2)} C) { (0,0), (1,2), (2,1), (1,1)} D) {0,1,2} Question 2 Which of the following is a nontrivial subgroup of the quaternion group of order 8. This group is plus or minus I, J, K, L where signs multiply as usual and I is a multiplicative identity, the squares of J,K,L are -I, and JK=L, KJ=-L, KL=J,LK=-J,LJ=K,JL=-K. The subgroup in question is cyclic. A) { J,K,L } B) {I,J,K,L} C) {I,J} D) I,J,-I,-J Question 3 Which of the following is a nontrivial subgroup of the symmetric group of degree 3 consisting of all permutations of the numbers 1,2,3? The element e denotes the identity permutation and other permutations are written in cycle form. This subgroup will be cyclic A) { e, (1,2,3)} B) {e,(1,2) } C) { e, (1,2), (1,2,3) } D) { e, (1,2) ,(1,3} Question 4 Consider the proof that the composition f(g(x)) of two one-to-one functions g: A to B and f:B to C is one-to-one. Suppose that it starts by assuming f(g(x))=f(g(y)). What is a valid and nontrivial second step in the proof? A) Because f is onto, g(x)=g(y) B) By assumption x=y C) Because f is one-to-one, g(x)=g(y) D) x and y are never equal
Which of the following is a nontrivial subgroup of where is the additive group of integers modulo 3 consisting of { 0,1,2 }, and the Cartesian product is the set of ordered pairs?
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A) { (0,0), (1,1), (2,2) }
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B) { (0,0) , (1,2)}
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C) { (0,0), (1,2), (2,1), (1,1)}
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D) {0,1,2}
Question 2Which of the following is a nontrivial subgroup of the quaternion group of order 8. This group is plus or minus I, J, K, L where signs multiply as usual and I is a multiplicative identity, the squares of J,K,L are -I, and JK=L, KJ=-L, KL=J,LK=-J,LJ=K,JL=-K. The subgroup in question is cyclic.
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A) { J,K,L }
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B) {I,J,K,L}
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C) {I,J}
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D) I,J,-I,-J
Question 3Which of the following is a nontrivial subgroup of the symmetric group of degree 3 consisting of all permutations of the numbers 1,2,3? The element e denotes the identity permutation and other permutations are written in cycle form. This subgroup will be cyclic
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A) { e, (1,2,3)}
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B) {e,(1,2) }
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C) { e, (1,2), (1,2,3) }
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D) { e, (1,2) ,(1,3}
Question 4Consider the proof that the composition f(g(x)) of two one-to-one functions g: A to B and f:B to C is one-to-one. Suppose that it starts by assuming f(g(x))=f(g(y)). What is a valid and nontrivial second step in the proof?
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A) Because f is onto, g(x)=g(y)
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B) By assumption x=y
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C) Because f is one-to-one, g(x)=g(y)
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D) x and y are never equal
Question 5For the following two functions, which of the following is correct: f(x)=x4 +1 from the set of all real numbers to itself, g(x)={ (1,1), (2,3), (3,2)} from the set {1,2,3} to itself?
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A) both are one-to-one
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B) the first, but not the second, is one-to-one
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C) the second, but not the first, is one-to-one
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D) neither is one-to-one
Question 6For the following two functions, which of the following is correct: f(x)=x4 +1 from the set of all real numbers to itself, g(x)={ (1,1),(2,3),(3,2)} from the set {1,2,3} to itself?
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A) both are onto
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B) the first, but not the second, is onto
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C) the second, but not the first, is onto
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D) neither is onto
Question 7Which of the following is always true of composition of functions, whether are not they are one-to-one or onto?
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A) it is associative
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B) it is commutative
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C) it is constant
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D) the composition is invertible
Question 8Compute the composition (product) of these two permutations
( 1 2 3 4 5 6 ) ( 3 5 4 1 2 6 ) ,
( 1 2 3 4 5 6 ) ( 2 4 5 3 1 6 ) Is it, in some order (we use the order of the book)
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A)
( 1 2 3 4 5 6 ) ( 1 2 3 4 5 6 ) -
B)
( 1 2 3 4 5 6 ) ( 5 1 2 4 3 6 ) -
C)
( 1 2 3 4 5 6 ) ( 6 1 2 4 3 5 ) -
D)
( 1 2 3 4 5 6 ) ( 2 4 5 3 1 6 ) Question 9
Factor this permutation into cycles:
( 1 2 3 4 5 6 ) ( 2 4 5 3 1 6 ) . This is not in cycle form. However the 4 possible answers are in cycle form.
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A) (1,2,3,4,5,6)
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B) (1,5,4,3,2,6)
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C) (1,2,4,3,5)
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D) (1,2,3)(4,5,6)
Question 10Suppose a square is represented with these 4 vertices in order: A is (1,1), B is (1,-1), C is (-1,-1), D is (-1,1). Note that these go clockwise around the square. Which of the following is a valid symmetry and a valid classification of that symmetry?
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A) A,B,C,D in order go to B,C,D,A a rotation by 90 degrees
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B) A,B,C,D in order go to D,C,A,B a reflection
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C) A,B,C,D in order go to D,C,B,A a rotation by 90 degrees
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D) A,B,C,D in order go to A,B,D,C a reflection
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