Discrete Mathematics Proofs for ∃ and ∀

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?

• You gave the solution. There really is no other one in such a simple case. Oct 8, 2018 at 17:21
• This seems like a math question, so off-topic here. Oct 8, 2018 at 18:39
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– Raphael
Oct 9, 2018 at 20:39
• 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.
– Raphael
Oct 9, 2018 at 20:39
• @YuvalFilmus We usually treat logics questions as ontopic, why not this one?
– Raphael
Oct 9, 2018 at 20:40