Question 1Which 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 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. A) { J,K,L } B) {I,J,K,L} C) {I,J} D) I,J,-I,-JQuestion 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 A) { e, (1,2,3)} B) {e,(1,2) } C) { e, (1,2), (1,2,3) } 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? 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 equalQuestion 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? A) both are one-to-one B) the first, but not the second, is one-to-one C) the second, but not the first, is one-to-one D) neither is one-to-oneQuestion 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? A) both are onto B) the first, but not the second, is onto C) the second, but not the first, is onto D) neither is ontoQuestion 7Which of the following is always true of composition of functions, whether are not they are one-to-one or onto? A) it is associative B) it is commutative C) it is constant D) the composition is invertibleQuestion 8Compute the composition (product) of these two permutations(123456)(354126),(123456)(245316)Is it, in some order (we use the order of the book) A)(123456)(123456) B)(123456)(512436) C)(123456)(612435) D)(123456)(245316)Question 9Factor this permutation into cycles:(123456)(245316). This is not in cycle form. However the 4 possible answers are in cycle form. A) (1,2,3,4,5,6) B) (1,5,4,3,2,6) C) (1,2,4,3,5) 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? A) A,B,C,D in order go to B,C,D,A a rotation by 90 degrees B) A,B,C,D in order go to D,C,A,B a reflection C) A,B,C,D in order go to D,C,B,A a rotation by 90 degrees D) A,B,C,D in order go to A,B,D,C a reflection

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Question 1Which 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 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. A) { J,K,L } B) {I,J,K,L} C) {I,J} D) I,J,-I,-JQuestion 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 A) { e, (1,2,3)} B) {e,(1,2) } C) { e, (1,2), (1,2,3) } 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? 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 equalQuestion 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? A) both are one-to-one B) the first, but not the second, is one-to-one C) the second, but not the first, is one-to-one D) neither is one-to-oneQuestion 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? A) both are onto B) the first, but not the second, is onto C) the second, but not the first, is onto D) neither is ontoQuestion 7Which of the following is always true of composition of functions, whether are not they are one-to-one or onto? A) it is associative B) it is commutative C) it is constant D) the composition is invertibleQuestion 8Compute the composition (product) of these two permutations(123456)(354126),(123456)(245316)Is it, in some order (we use the order of the book) A)(123456)(123456) B)(123456)(512436) C)(123456)(612435) D)(123456)(245316)Question 9Factor this permutation into cycles:(123456)(245316). This is not in cycle form. However the 4 possible answers are in cycle form. A) (1,2,3,4,5,6) B) (1,5,4,3,2,6) C) (1,2,4,3,5) 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? A) A,B,C,D in order go to B,C,D,A a rotation by 90 degrees B) A,B,C,D in order go to D,C,A,B a reflection C) A,B,C,D in order go to D,C,B,A a rotation by 90 degrees D) A,B,C,D in order go to A,B,D,C a reflection

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