First Order Logic: ∀x,y a(x) ∧ a(y)

[ver_mathstats]

Homework Statement

Let sigma = {a,b,c} be a finite alphabet. Construct formulas that specify the following languages over sigma.

Relevant Equations

sigma = {a,b,c}

An example we were given is as follows: {ua|u∈∑*} (where ∑* is set of all words over ∑) so we have ∀x. last(x) → a(x).

I am given {awa|w∈∑*} to do, and I know that I have to express that a is the first letter and last letter in a word. Could I write it as:
∀x,y ( a(x) ∧ a(y) ∧ x<y → ∃z(x<z<y)), am I on the right track with this, any help would be appreciated, thank you.

 

[mighty2000]

Hello,

Yes, you are on the right track with your expression. However, there are a few minor changes that can be made to make it more accurate.

Firstly, the expression should be ∀x,y ( a(x) ∧ a(y) ∧ x<y → ∃z(x<z ∧ z<y)). This ensures that z is between x and y, rather than just being larger than x and smaller than y.

Additionally, you can add in the condition that x and y are words in the set ∑*, so the final expression would be: ∀x,y ( a(x) ∧ a(y) ∧ x<y ∧ x∈∑* ∧ y∈∑* → ∃z(x<z ∧ z<y)).

Hope this helps! Let me know if you have any further questions. Good luck with your work.