Closure properties of finite state transducers Given $T_1, T_2\colon \Sigma^* \to \Gamma^*$ ($\Gamma$ is output alphabet), let $\Delta(T_1, T_2)$ 
      consist of all input strings $w \in Σ^*$ where $T_1(w) \neq T_2(w)$. 
      Prove that FSTs are not closed under this operation $\Delta(T_1,T_2)$.


What I have tried is to create two FSTs $T_1, T_2$ (as 5 tuple functions of form  $(Q, \Sigma, \Gamma, \delta, s, γ)$  where $\Sigma$ - input symbols, $\Gamma$ - output symbols, $δ\colon Q \times \Sigma → Q$ transition function, to prove that the language $S = \{ w \mid T_1(w) \neq T_2(w) \}$ is not regular. I am having trouble creating the two FSTs. Especially defining the transition functions for the two FSTs. Any help with this is appreciated.Pretty much what I am trying to prove is there are two FST's $T_1$ and $T_2$ such that the language $L = \{w \mid T_1(w) \ne T_2(w)\}$ is not regular. To prove this, what I want to do is create the two FST's $T_1$ and $T_2$. I am not sure on how to define a 6-tuple FST. Especially coming up with the transition functions for the two FSTs
It is probably a better idea first to concentrate on the complement, i.e., the equality language $\{ w\in \Sigma^* \mid T_1(w) = T_2(w) \}$.

Also, simple FST's will do, as even for morphisms $h_i: \{a,b\} \to \{1\}^*$, $\{ x\in \{a,b\}^* \mid h_1(x) = h_2(x) \}$ need not be regular.

And I want to add that it has been shown that $\Delta(T_1,T_2)$ is always context-free.