exercises 1 final

EXERCISES FUNCTIONS (MAPPINGS) 1. Let S={ w,x,y,z } and T={1 , 2 , 3 , 4 }, and define α : S→T and β : S→T by α ( w )=2, α ( x )=4, α ( y )=1, α ( z) =2 and β ( w ) =4, β ( x) =2, β ( y) =3, β ( z) =1. a. Is α one-to-one? Is α onto? b. Is β one-to-one? Is β onto? 2. Let α,β,∧γ be mappings from Z to Z defined by α ( n) =2 n, β ( n) =n+ 1, and γ ( n) =n 3 for each n∈Z. a. Which of the three mappings are onto? b. Which of the three mappings are one-to-one? 3. Each f defines a mapping from R (or a subset of R) to R. Determine which of these mappings are onto and which are one-to-one. a. f ( x )=2 x b. f ( x )=x−4 c. f ( x )=x 3 d. f ( x )=x 2 +x e. f ( x )=e x f. f ( x )=tan x EQUIVALENCE RELATIONS 1. Define a relation ~ on R by ab iff | a| = |b |. Prove that ~ is an equivalence relation on R. 2. Define a relation ≈ on the set N of the natural numbers by a≈b iff a=b∙ 10 k for some k∈Z. Prove that ≈~ is an equivalence relation on N. BINARY OPERATIONS 1. Determine whether or not ¿ is a binary operation on the given set. a. a∗b= a+b ab , on the set Z b. a∗b=a ln b, on the set R +¿¿ c. a∗b= | a−b|, on the set Z* d. a∗b= √ | ab |, on the set Q 2. Which of the following equations define operations on the set of integers? Of those that do, which are associative? Which are commutative? Which have identity elements? a. m∗n= m+ n 2 b. m∗n=m c. m∗n=mn 2

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abstract algebra

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EXERCISESFUNCTIONS (MAPPINGS)1. Let and , and define and by , , , and , , , a. Is one-to-one? Is onto? b. Is one-to-one? Is onto?

2. Let be mappings from Z to Z defined by , , and for each a. Which of the three mappings are onto?b. Which of the three mappings are one-to-one?

3. Each defines a mapping from R (or a subset of R) to R. Determine which of these mappings are onto and which are one-to-one.a. b. c. d. e. f. g.

EQUIVALENCE RELATIONS1. Define a relation ~ on R by iff Prove that ~ is an equivalence relation on R. 2. Define a relation on the set N of the natural numbers by iff for some Prove that ~ is an equivalence relation on N.

BINARY OPERATIONS1. Determine whether or not is a binary operation on the given set.a. b. on the set Zc. , on the set d. , on the set Z*e. , on the set Q

2. Which of the following equations define operations on the set of integers? Of those that do, which are associative? Which are commutative? Which have identity elements?a. b. c. d. e. f. g. h. i.

3. Each of the following is a binary operation on R. Indicate whether or noti. it is commutativeii. it is associativeiii. R has an identity element with respect to iv. every has an inverse with respect to whenever identity existsa. b. c. d. e. f. g. h. i.

MAGMAS ISOMORPHISM1. Describe an isomorphism between the sets defined on the given operations by the ff: 123

1123

2231and

3312

2. Determine whether the given mapping is an isomorphism of the first magma with the second. If it is not an isomorphism, why not?a. with where is the determinant of matrix Ab. with where is the determinant of matrix Ac. with where for

PERMUTATIONS1. Compute each of the following products (or the operation ) in . Write your answer as a single permutation.a. b. (145)(37)(682)c. (17)(628)(9354)d. (71825)(36)(49)e. (12)(347)f. (147)(1678)(74132)2.

3. Write each of the following permutations in as a product of disjoint cycles.a. b. c. d. e. 4.

5. Express each of the following as a product of transpositions in .a. b. (137428)c. (416)(8235)d. (123)(456)(1574)e. 6.

7. Determine which of the following permutations is even, and which is odd.a. b. c. d. (12)(76)(345)e. (1276)(3241)(7812)f. (123)(2345)(1357)

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