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Elements Of Modern Algebra

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.4: Binary Operations

Problem 6E

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Question

Answer if TRUE or FALSE

Let S be the set of integers and H be the set of all odd integers. Then the subset H of S is
closed under the usual multiplication.
True
False
Let a + b = ab – 2. Then 2 is the identity element of Z under *.
True
False
Let a + b = ab – 2. Then the inverse element of a in Z does not exist.
True
False

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In each part following, a rule that determines a binary operation on the set of all integers is given. Determine in each case whether the operation is commutative or associative and whether there is an identity element. Also find the inverse of each invertible element. b. d. f. h. j. l. for n. for
Prove that if and are integers such that and , then either or .
Let a,b,c and d be integers such that ab and cd. Prove that acbd.
Let a be an integer. Prove that 3|a(a+1)(a+2). (Hint: Consider three cases.)
25. Prove that if and are integers and, then either or. (Hint: If, then either or, and similarly for. Consider for the various causes.)
Prove that a nonzero element in is a zero divisor if and only if and are not relatively prime.
Let a and b be integers such that ab and ba. Prove that b=0.
Let be the set of all elements of that have one row that consists of zeros and one row of the form with . Show that is closed under multiplication in . Show that for each in there is an element in such that . Show that does not have an identity element with respect to multiplication.
True or False Label each of the following statements as either true or false. Every permutation has an inverse.
26. Let be an integer. Prove that . (Hint: Consider two cases.)
Let a be an odd integer. Prove that 8|(a21).
True or False Label each of the following statements as either true or false. 5. Let be a set of integers closed under subtraction. If and are elements of , then is in for any in .

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