Relation between homotopy of pairs and homotopy relative.

Question

Let $(X,A)$ and $(Y,B)$ be CW-pairs, and let $f,g:(X,A)⟶(Y,B)$ be maps of pairs.

Some definitions:

1. A homotopy relative to $A ⊆ X$ from $f$ to $g$ is defined to be a map $F:X×I⟶Y$ such that $F(x,0)=f(x),F(x,1)=g(x)$ and $F(A,t)=f(A)=g(A)$ for all $t$.
2. A homotopy of pairs from $f$ to $g$ is defined to be a map $F:X×I⟶Y$ such that $F(x,0)=f(x),F(x,1)=g(x)$ and $F(A,t)⊆B$ for all $t$.

Question: Now from 1. we obtain $F(A,t)=f(A)=g(A) ⊆ B$ for all $t$ which satisifies 2.. Then isn't a homotopy relative to $A$ just a homotopy of pairs?

Answer 1

If $f, g : (X,A) \to (Y,B)$ are maps of pairs and $F : X \to Y$ is a homotopy from $f$ to $g$ rel. $A$, then $F$ is indeed a homotopy of pairs from $f$ to $g$. However, in general there are more homotopies of pairs than homotopies rel. $A$.