1. Complete the truth table for the following logical expression. You may not use any logical equivalences to simplify the expression. Be sure to include a column for each subexpression you evaluate.

((( )) ( ))

2. Determine whether the following expression is a tautology, a contingency, or a contradiction by using the logical equivalences .

(( ) (( ) ( )))

3. Let the universe of discourse be all animals. Specify what variables you are using for each ATOMIC predicate, and then translate the following statements into predicate logic expressions.

a. At least one bird can fly and swim.

b. Rats cannot fly.

4. Determine if the following logical and arithmetic statement is true or false and justify your answer.

if (19 + 1) > (25 - 4) or ( 11 + 3 ) > 12 then 12 < 15 or 14 < (20 - 9)

5. State the inverse, converse, and contrapositive of the following implication expression as English sentences. Ensure that you list the symbols you will use for each ATOMIC predicate and simplify your expressions and distribute your negation operations as much as possible before translating.

If it is entertaining then it is not impossible and it is not difficult