0.4 Equivalences
1.
a. It is an equivalence by Example 4.
c. Not an equivalence. x ≡ x only if x = 1, so the reflexive property fails.
e. Not an equivalence. 1 ≡ 2 but 26 ≡ 1, so the symmetric property fails.
g. Not an equivalence. x ≡ x is never true. Note that the transitive property also fails.
i. It is an equivalence by Example 4. [(a, b)] = {(x, y) 
y − 3x = b − 3a} is the line with slope 3 through (a, b).
y − 3x = b − 3a} is the line with slope 3 through (a, b).
2. In every case (a, b) ≡ (a1, b1) if α(a, b) = α(a1, b1) for an appropriate function 
. Hence ≡ is the kernel equivalence of α.
. Hence ≡ is the kernel equivalence of α.
a. The classes are indexed by the possible sums of elements of U.
c. The classes are indexed by the first components.
3.
a. It is the kernel equivalence of 
where α(n) = n2. Here [n] = { − n, n} for each n. Define 
by σ[n] = |n|, where |n| is the ...
where α(n) = n2. Here [n] = { − n, n} for each n. Define by σ[n] = |n|, where |n| is the ...Get Introduction to Abstract Algebra, Solutions Manual, 4th Edition now with the O’Reilly learning platform.
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