Answer
Let p represent "it walks like a duck."
Let q represent "it talks like a duck."
Let r represent "it is a duck."
"If it walks like a duck and it talks like a duck, then it is a duck" in symbolic form is p $\land$ q $\rightarrow$ r.
"Either it does not walk like a duck or it does not talk like a duck, or it is a duck" is (~p $\lor$~q) $\lor$r.
"If it does not walk like a duck and it does not talk like a duck, then it is not a duck" is ~p $\land$ ~q $\rightarrow$ ~r.
The first two sentences are logically equivalent, p $\land$ q $\rightarrow$ r $\equiv$ (~p $\lor$~q) $\lor$r. The last sentence is not logically equivalent to either of the first two. See the truth table.
Work Step by Step
To construct the truth table, first fill in the 8 possible combinations of truth values for p, q, and r. To evaluate p ∧ q recall the definition of AND (a ∧ b is true when, and only when, both a and b are true. If either a or b is false or if both are false, a ∧ b is false). To evaluate p ∧ q → r, recall by the definition of a conditional statement, when the if element is T and the then element is F, the statement is F. In all other cases the statement is T. To evaluate ~p ∨ ~q ∨ r recall the definition of OR (a ∨ b is true when either a is true, or b is true, or both a and b are true; it is false only when both a and b are false). The two statements are only logically equivalent if, and only if, they have identical truth values for each possible substitution of statements for their statement variables. The truth table shows p $\land$ q $\rightarrow$ r $\equiv$ (~p $\lor$~q) $\lor$r.
To evaluate the truth table of ~p $\land$ ~q $\rightarrow$ ~r recall the definition of AND and the definition of a conditional statement (restated above). The truth table shows the truth values of ~p $\land$ ~q $\rightarrow$ ~r are not the same as the other two statements.
