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Each of the following pairs is composed of a non-empty set M and some binary operation ° on this set. In which case does the set M not contain a neutral element for the operation °? (a) M = R[x]; p°q = p △q (polynomial composition). (b) M = R[x]; p °q = pq (usual polynomial multiplication). (c) M = {(a,b) ∈ R^2 | a + b = 1 }; (a,b) °(c,d) = (a+c,b+d) (usual addition of ordered pairs).

Each of the following pairs is composed of a non-empty set M and some binary operation ° on this set. In which case does the set M not contain a neutral element for the operation °? (a) M = R[x]; p°q = p △q (polynomial composition). (b) M = R[x]; p °q = pq (usual polynomial multiplication). (c) M = {(a,b) ∈ R^2 | a + b = 1 }; (a,b) °(c,d) = (a+c,b+d) (usual addition of ordered pairs).

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.3: Zeros Of Polynomials
Problem 32E
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Each of the following pairs is composed of a non-empty set M and some binary operation ° on this set. In which case does the set M not contain a neutral element for the operation °?

(a) M = R[x]; p°q = p △q (polynomial composition).

(b) M = R[x]; p °q = pq (usual polynomial multiplication).

(c) M = {(a,b) ∈ R^2 | a + b = 1 }; (a,b) °(c,d) = (a+c,b+d) (usual addition of ordered pairs).

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