I'm a bit confused on how to use homomorphims to prove irregularity or to prove that a language is not context free. This is what I'm currently thinking: Example 1:Let $L = \{ a^{i}b^{j}c^{k} : i = j = k \}$ and$h: \{a, b, c\} \rightarrow \{a, b, \epsilon\}$ be defined as follows:$$h(x) = \begin{cases} \epsilon & x = c\\ x & o.w.\end{cases}$$Then it follows that$h(L) = \{a^ib^j: i=j \} = \{a^{n}b^{n}: n \ge 0 \}$.Now because $\{a^nb^n: n \ge 0 \}$ is known to be irregular, then so is $L$.Example 2:We know that $P = \{0^p : p \text{ is a primes}\}$ is irregular (by the pumping lemma). Defined $h: \{0\} \to \{1\}$ as $h(1)=0$.It follows that $h(P) =  ${1^p : p \text{ is a primes}}$ and therefore is irregular.I wrote these as answers to an exercise but I've been told that they are incorrect. I don't understand why?
How to use homomorphisms to prove irregularity
formal languages;regular languages;check my answer