Is a string with the null character the same as the same string without the null character? Is this { Λ, 1 } the same as this { 1 }? 
For instance if you have the following grammar:
S -> 0X|Y1
X -> 1|Λ
Y -> 0|Λ
Will every string created by it be ambiguous?

The set $\{\Lambda,1\}$ has two elements, the empty string and the string $1$; the set $\{1\}$ has only one element, the string $1$. Thus, the two sets are not equal.
Your grammar can generate three different strings, $0,1$, and $01$. The strings $0$ and $1$ have unique derivations,
$$S\Rightarrow 0X\Rightarrow 0$$
and
$$S\Rightarrow Y1\Rightarrow 1\;.$$
The string $01$, however, does have two derivations,
$$S\Rightarrow 0X\Rightarrow 01$$
and
$$S\Rightarrow Y1\Rightarrow 01\;.$$
Both are leftmost derivations, so the grammar is ambiguous. The language $\{0,1,01\}$ itself is not inherently ambiguous, however: you can easily write an unambiguous grammar for it. (Note that ambiguity is not a property of individual strings: it’s a property of grammars and languages.)
