Alon,

I usually use the function Equivalent together with TautologyQ to test if two logical expressions are equivalent, like this:

(* Let us prove that ((A OR B) => C) <=> (A => C) AND (B => C) *)
Equivalent[Implies[Or[A,B],C], And[Implies[A,C],Implies[B,C]]]//TautologyQ

Now, if you substitute your own expression instead of these two simple ones shown above, then it should work. Note that this is for Mathematica and not for Wolfram Alpha (I don't know if Wolfram Alpha understands the same syntax or not).

I fixed the first link... sorry about that. The second one discusses the area of automated theorem proving--often (I think) it's in the realm of automatically generating a boolean proof (via manipulating logical expressions) of a logical equivalence. Which is that I think you are wanting to do.

You might be interested in this:

http://www.wolframscience.com/nksonline/page-1157c-text

and this

http://en.wikipedia.org/wiki/Automated_theorem_proving

Is you + sign the same as Or and your * sign the same as And? I believe so. So with this the two "sentences" corresponding to your two logial expressions would be (if I haven't made any mistakes) respectively:

(a and c) or (c and d) or (a and b and d) or (!a and !b and d) or (a and !b and !d) or (!a and !c and !d)

and

(a or !c or d) and (!a or !b or c or d) and (a or !b or c or !d) and (!a or b or c or !d)

As Mathematica expressions these are respectively:

expression1=(a && c) || (c && d) || (a && b && d) || ( ! a &&  ! b &&  d) || (a &&  ! b &&  ! d) || ( ! a &&  ! c &&  ! d)

and

 expression2= (a ||  ! c || d) && ( ! a ||  ! b || c || d) && (a ||  ! b || c ||  ! d) && ( ! a || b || c ||  ! d)

And we can see if these are the sam by using BooleanTable:

In[14]:= BooleanTable[expression1, {a, b, c, d}]

Out[14]= {True, True, True, False, True, True, False, True, True, \
False, False, True, True, False, True, True}

In[15]:= BooleanTable[expression2, {a, b, c, d}]

Out[15]= {True, True, True, False, True, True, False, True, True, \
False, False, True, True, False, True, True}

In[16]:= BooleanTable[expression2, {a, b, c, d}] === 
 BooleanTable[expression1, {a, b, c, d}]

Out[16]= True

So, what about Wolfram Alpha? One problem is that forming the And of the two word expressions is too long for the Wolfram Alpha input field... so it goes nowhere.

One possibility is to do two Wolfram Alpha calls

((a and c) or (c and d) or (a and b and d) or (!a and !b and d) or (a and !b and !d) or (!a and !c and !d)) truth table

and

((a or !c or d) and (!a or !b or c or d) and (a or !b or c or !d) and (!a or b or c or !d)) truth table

and visually compare the truth tables.

perhaps there are some other approaches in pure Wolfram Alpha that others can suggest.

What do you mean by Wolfram: Mathematica or Wolfram Alpha?

Also, what do you mean by Negate? Do you mean logical negation or changing the sign of an algebraic expression?

I think you are meaning logical operations. If so, read the documentation (always read the documentation). Here are some links if you are meaning Mathematica.

http://reference.wolfram.com/language/guide/LogicAndBooleanAlgebra.html

http://reference.wolfram.com/language/howto/UseLogicalOperators.html

http://reference.wolfram.com/language/guide/TheoremProving.html