Answer:
If it walks like a duck and it talks like a duck, then it is a duck.
and
Either it does not walk like a duck or it does not talk like a duck, or it is a duck.
are logically equivalent to each other.
but neither of the two is logically equivalent to
If it does not walk like a duck and it does not talk like a duck, then it is not a duck.
Explanation:
Given statements:
Statemen 1:
If it walks like a duck and it talks like a duck, then it is a duck.
Statement 2:
Either it does not walk like a duck or it does not talk like a duck, or it is a duck.
Statement 3:
If it does not walk like a duck and it does not talk like a duck, then it is not a duck.
Let
- p be the statement: it walks like a duck
- q be the statement: it talks like a duck
- r be the statement: it is a duck
Using p = it walks like a duck , q =it talks like a duck, r = it is a duck the given statements can be written in symbolic form as:
Statement 1:
p ∧ q → r
The ∧ symbol shows that if both p and q are true then, they imply r. This means both p and q together imply r
Statement 2:
~p ∨ ~q ∨ r
Here the statement p and q are negated and joined using or. So either negation p or negation of q or r (alternative)
Statement 3:
~p ∧ ~q → ~r
The ∧ symbol shows that if both negation of p and negation of q are true then, they imply r. This means both negated p and negated q together imply negated r
Statement 1:
p ∧ q → r ≡ ~(p∨q) ∨ r Using conditional equivalence p→q ≡ ~p ∨ q
≡ (~p ∧ ~q ) ∨ r
You can see it is equivalent to Statement 2 i.e. ~p ∧ ~q → ~r
Hence Statement 1 and Statement 2 are logically equivalent.
Now Statement 3:
~p ∧ ~q → ~r ≡ ~(~p ∧ ~q ) ∨ ~r Using conditional equivalence p→q ≡ ~p ∨ q
≡ ~(~p ) ∨ ~(~q ) ∨ ~r Using De Morgan's Law ~(p∧q) ≡ ~p ∨~q
≡ p ∨ q ∨ ~r Using Double Negation Law ~(~p)≡p
This shows that Statement 3 is neither logically equivalent to Statement 1 nor logically equivalent to Statement 2.
Proof by truth table is attached. The table shows that the columns for Statement 1 and Statement 2 have same truth values.
Hence
"If it walks like a duck, and it talks like a duck, then it is a duck,"
and
"If it does not walk like a duck, and does not talk like a duck, then it is not a duck,"
are logically equivalent.
The table also shows that column for Statement 3 does not match with either of the columns for Statement 1 and Statement 2. So
If it does not walk like a duck and it does not talk like a duck, then it is not a duck.
is not logically equivalent to Statement 1 and Statement 2.