# Question about logic and recurrence relation

1. Sep 28, 2010

### johnsmiths

1. The problem statement, all variables and given/known data
find a logical expression using only ∧ and ¬ operators which is logically equivalent to (p ∨ q)

3. The attempt at a solution
losing direction
what should I first consider?

There is another question about recurrent relation.

Suppose that a mathematical expression can only be formed by the following symbols: 0, 1,
2, …, 9, ×, +, /. Some examples are “0 + 9”; “2 + 2 × 8”; “1 / 5 + 6”. Let an be the the number
of such mathematical expression of length n (e.g. “0 + 9” is considered of length 3). Find a
recurrence relation for an and compute the closed form for an.
[Some clarification: We define a number as follows
- 0, 1, 2, …, 9 is a number
- If x is a number, then x0, x1, …, x9 is a number
We define a valid expression as follows
- E is a valid expression if E is a number
- If E, F are valid expressions, then E + F, E × F, E / F are also valid expressions.
For example: 1+50/4 is an expression of length 6, and 09×00/5 is an expression of length 7.]

2. Sep 28, 2010

### fss

You should consider what it means to be logically equivalent, then look up "tautologies."

3. Sep 28, 2010

### johnsmiths

What do you mean is finding something $$\rightarrow$$ ( p$$\vee$$q ) and make it a tautology?

4. Sep 28, 2010

### fss

I do not understand what the above question is asking.

Do you understand what it means when two statements are logically equivalent?

5. Sep 28, 2010

### johnsmiths

It means p$$\leftrightarrow$$q is a tautology.

Maybe I make some mistakes with my question. And after I know that, what should I do next?
Just guessing or there is a systematic way to work out the solution.

6. Sep 28, 2010

### fss

It's not guessing, but it's not really "systematic" because it's a very simple question. All you need to do is use what you know about tautologies to construct an equivalent statement. The original statement p v q has a very simple logical equivalent that you should be able to deduce with only a little thinking once you find the proper tautological statement.