Homotopy group of pairs: equivalent descriptions
Solution 1:
This only makes sense if $*\in A$. Then take the basepoint of $P(X,A)$ to be the constant path mapping $I\mapsto \{*\}$. Now an element of $\pi_{n-1}(P(X,A))$ is a map (up to homotopy) $I^{n-1}\to P(X,A)$, sending $\partial I^{n-1}$ to the constant map.
Equivalently, it is a map (up to homotopy), $I^{n-1}\times I \to X$, which:
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Maps $I^{n-1} \times \{0\}$ to $*$, as every path starts at $*$.
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Maps $\partial I^{n-1}\times I$ to $*$, as the basepoint of $P(X,A)$ is the constant map to $*$.
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Maps $I^{n-1} \times \{1\}$ to $A$, as paths end in $A$.