Let T be the set of real numbers. Consider T as an algebraic structure with addition operation and multiplication operation given by x + y = min{x,y}, x©y=x+y - 2. Name the identity element in T for the operation, and prove the inverse law for O. 2 Prove the distributive law in T. [Hint: consider two cases x ≤ y, x > y.] 3 Prove that the set T with addition and multiplication Ⓒ is not a ring.

Let T be the set of real numbers. Consider T as an algebraic structure with addition operation and multiplication operation given by x + y = min{x,y}, x©y=x+y - 2. Name the identity element in T for the operation, and prove the inverse law for O. 2 Prove the distributive law in T. [Hint: consider two cases x ≤ y, x > y.] 3 Prove that the set T with addition and multiplication Ⓒ is not a ring.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.4: Binary Operations

Problem 9E: 9. The definition of an even integer was stated in Section 1.2. Prove or disprove that the set of...

See similar textbooks

Similar questions

15. Let be a binary operation on the non empty set . Prove that if contains an identity element with respect to , the identity element is unique.
Let a and b be constant integers with a0, and let the mapping f:ZZ be defined by f(x)=ax+b. Prove that f is one-to-one. Prove that f is onto if and only if a=1 or a=1.
Let R be the relation defined on the set of integers by aRb if and only if ab. Prove or disprove that R is an equivalence relation.
9. The definition of an even integer was stated in Section 1.2. Prove or disprove that the set of all even integers is closed with respect to a. addition defined on . b. multiplication defined on .
21. A relation on a nonempty set is called irreflexive if for all. Which of the relations in Exercise 2 are irreflexive? 2. In each of the following parts, a relation is defined on the set of all integers. Determine in each case whether or not is reflexive, symmetric, or transitive. Justify your answers. a. if and only if b. if and only if c. if and only if for some in . d. if and only if e. if and only if f. if and only if g. if and only if h. if and only if i. if and only if j. if and only if. k. if and only if.
13. Let Z denote the set of all integers, and let Prove that .
[Type here] 7. Let be the set of all ordered pairs of integers and . Equality, addition, and multiplication are defined as follows: if and only if and in , Given that is a ring, determine whether is commutative and whether has a unity. Justify your decisions. [Type here]
2. Prove that is commutative if and only if is commutative.
Let g:AB and f:BC. Prove that f is onto if fg is onto.
True or False Label each of the following statements as either true or false. 6. Every bijection is both one-to-one and onto.
43. Let the operation of addition be as defined in Exercise 42. Prove each of the following statements. a. b. 42. Let the operation of addition be defined on subsets and of by
Let be as described in the proof of Theorem. Give a specific example of a positive element of .