# The Exact Homology Sequence (Exact Sequence of Triples)

Problem:

Let X = X_1 / X_2, and A = X_1 / X_2. Using the exact sequence of triples, show that if the inclusion (X_1, A) --> (X, X_2) induces an isomorphism on homology, then the same holds for the inclusion (X_2, A) --> (X, X_1).

Notation:

X_1 is X subscript 1

/ is union

/ is intersection

--> is an inclusion map

H_q (X, A) is the quotient module, the qth relative homology module of X mod A

Need a step-by-step proof outline.

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#### Solution Summary

This solution is comprised of a detailed explanation to provide a step-by-step proof outline.

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