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Premises or Givens:

  • $∃x(A(x) → B(x))$
  • $∀x (B(x) → K(x))$

To Prove:

  • $∃x(A(x) → K(x))$

My Solution:

  1. $A(z) → B(z)$ From premise and Existential instantiation $x$ for $z$

  2. $B(z) → K(z)$ From premise and Universal instantiation $x$ for $z$

  3. $A(z) → K(z)$ Transitivity of 1,2

  4. $∃x(A(x) → K(x))$ From Existential generalization (Substitute $z$ for $x$)

OR

I was thinking about assuming $A(z)$ and then using Modus Ponens to get $B(z)$ and then further $K(z)$, then using the deduction theorem on $A(z)$ and $K(z)$, and then using Existential Generalization on that statement by substituting $x$ for $z$.

Can someone suggest which way would be more effective?

Is there any other effective way to solve it?

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  • $\begingroup$ You gave the solution. There really is no other one in such a simple case. $\endgroup$ – Yuval Filmus Oct 8 '18 at 17:21
  • $\begingroup$ This seems like a math question, so off-topic here. $\endgroup$ – Yuval Filmus Oct 8 '18 at 18:39
  • $\begingroup$ Welcome to Computer Science! The title you have chosen is not well suited to representing your question. Please take some time to improve it; we have collected some advice here. Thank you! $\endgroup$ – Raphael Oct 9 '18 at 20:39
  • $\begingroup$ We discourage "please check whether my answer is correct" questions, as only "yes/no" answers are possible, which won't help you or future visitors. See here and here. Can you edit your post to ask about a specific conceptual issue you're uncertain about? As a rule of thumb, a good conceptual question should be useful even to someone who isn't looking at the problem you happen to be working on. If you just need someone to check your work, you might seek out a friend, classmate, or teacher. $\endgroup$ – Raphael Oct 9 '18 at 20:39
  • $\begingroup$ @YuvalFilmus We usually treat logics questions as ontopic, why not this one? $\endgroup$ – Raphael Oct 9 '18 at 20:40
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Your solution is a good one, and that's how I'd prove it also. I don't believe a more straightforward or simpler version is possible.

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