1st Honours Algebra Quick Quiz 2 Click on the correct answer to the question. 1. Let X be a set. What does the notation 2X mean? A. X must be a number after all, and not a set. B. The set of all subsets of X. C. There's a typo. D. You can put an X just about anywhere and get away with it. 2. A bijection is another word for A. A permutation. B. An injection. C. A surjection. D. Two rejections. 3. Let f = (142)(3) and g = (12)(34). Then f o g is A. (134)(2) B. (13)(24) C. (1)(243) D. (1)(2)(3)(4) 4. The cardinality of the set S3 is A. 4 B. 1 C. 5 D. 3! 5. Let X be a set. In order for a relation on X to be considered an equivalence relation, what three properties must be satisfied? The relation must be A. Reflexive, symmetric and transitory B. Reflective, symmetric and transitive C. Reflexive, symmetric and transitive 6. The Fundamental Theorem of Equivalence Relations states that: If you have an equivalence relation on a set X, then the set P of equivalence classes forms of the set X. Moreover, if x and y are elements of X then x twiddles y if and only if the equivalence classes of x and y are . The missing underlined words are A. class, unequal B. a partition, equal C. an apparition, equivalent 7. Let P be a partition of a set X. We can define a relation on the set X which is an equivalence relation on X, as follows: Let x and y be elements of X then A. x twiddles y if x and y are both in the same subset of P. B. x twiddles y if y twiddles x. C. There's no such definition. D. x twiddles y if x and y like to have a good twiddle. 8. Complete the definition, where x and y are integers: If n divides x - y then x is said to be to y modulo n. A. equivalent B. equal C. congruent 9. An equivalence class of the integers under congruence is known as A. a modulo class B. a congruence class C. an integer class D. a boring class 10. Working modulo 11 it is easy to check that 45 is congruent to A. 5 B. 36 C. 1 D. 0 11. Working modulo 13 it is easy to check that 28 is congruent to A. -2 B. -15 C. -11 D. -8 12. Working modulo 6 it is easy to check that 2 × 21 is congruent to A. 2 B. 5 C. 0 D. 1 13. Does the equation 6x + 9y = 5 have integer solutions x and y? Why? A. Yes, because 3 doesn't divide 5 B. Yes, because 3 divides 5 C. No, because 3 doesn't divide 5 D. No, all solutions are rational. 14. Does the equation 42x + 343y = 21 have solutions x and y? Why? A. Yes, because 7 doesn't divide 21 B. Yes, because 7 divides 21 C. No, because 7 doesn't divide 21 D. No, all solutions are real. 15. The method for finding integer solutions x, y to an equation of the form ax + by = c is called A. The Division Algorithm B. The Euclidean Method C. The Division Method D. The Euclidean Algorithm If you think I've got an answer wrong, e-mail me at rbreams@vcu.edu so that I can correct it.
Click on the correct answer to the question.
1. Let X be a set. What does the notation 2X mean? A. X must be a number after all, and not a set. B. The set of all subsets of X. C. There's a typo. D. You can put an X just about anywhere and get away with it.
2. A bijection is another word for A. A permutation. B. An injection. C. A surjection. D. Two rejections.
3. Let f = (142)(3) and g = (12)(34). Then f o g is A. (134)(2) B. (13)(24) C. (1)(243) D. (1)(2)(3)(4)
4. The cardinality of the set S3 is A. 4 B. 1 C. 5 D. 3!
5. Let X be a set. In order for a relation on X to be considered an equivalence relation, what three properties must be satisfied? The relation must be A. Reflexive, symmetric and transitory B. Reflective, symmetric and transitive C. Reflexive, symmetric and transitive
6. The Fundamental Theorem of Equivalence Relations states that: If you have an equivalence relation on a set X, then the set P of equivalence classes forms of the set X. Moreover, if x and y are elements of X then x twiddles y if and only if the equivalence classes of x and y are . The missing underlined words are A. class, unequal B. a partition, equal C. an apparition, equivalent
7. Let P be a partition of a set X. We can define a relation on the set X which is an equivalence relation on X, as follows: Let x and y be elements of X then A. x twiddles y if x and y are both in the same subset of P. B. x twiddles y if y twiddles x. C. There's no such definition. D. x twiddles y if x and y like to have a good twiddle.
8. Complete the definition, where x and y are integers: If n divides x - y then x is said to be to y modulo n. A. equivalent B. equal C. congruent
9. An equivalence class of the integers under congruence is known as A. a modulo class B. a congruence class C. an integer class D. a boring class
10. Working modulo 11 it is easy to check that 45 is congruent to A. 5 B. 36 C. 1 D. 0
11. Working modulo 13 it is easy to check that 28 is congruent to A. -2 B. -15 C. -11 D. -8
12. Working modulo 6 it is easy to check that 2 × 21 is congruent to A. 2 B. 5 C. 0 D. 1
13. Does the equation 6x + 9y = 5 have integer solutions x and y? Why? A. Yes, because 3 doesn't divide 5 B. Yes, because 3 divides 5 C. No, because 3 doesn't divide 5 D. No, all solutions are rational.
14. Does the equation 42x + 343y = 21 have solutions x and y? Why? A. Yes, because 7 doesn't divide 21 B. Yes, because 7 divides 21 C. No, because 7 doesn't divide 21 D. No, all solutions are real.
15. The method for finding integer solutions x, y to an equation of the form ax + by = c is called A. The Division Algorithm B. The Euclidean Method C. The Division Method D. The Euclidean Algorithm