Homework #4:

Propositional-Logic Truth Tables

1. 1. Construct a truth table for the proposition:
      not(x and y)
      • Hint: To do this, first construct the truth table for x and y. Then add another column for its negation.

   2. Does this look like the truth table for any other propositional connective?

   ---

2. Construct the truth table for:
   (x or y) and not(x and y)

   ---

3. 
   "Exclusive disjunction", denoted by the connective "xor", can be defined as follows:
   "x xor y" is true if exactly one of x, y is true, and is false otherwise.

   Construct a truth table for "x xor y".

   ---

4. 1. Construct the truth table for:
      (not x) or y

   2. Does this look like the truth table for any other propositional connective?

   ---

5. 1. Construct the truth table for the "biconditional":
      x if and only if y

      where the biconditional of x and y is true if both of them have the same truth value, and false otherwise.

   2. Compare it to the truth table for exclusive disjunction.

   ---

6. Construct the truth table for:
   if [(if x, then y) and x], then y

   ---

7. Construct the truth table for:
   if [(if x, then y) and y], then x

   ---

8. Let's define a new connective, "nor", as follows:
   x nor y =def not(x or y)
   1. Construct the truth table for x nor y.
   2. Construct the truth table for x nor x.
   3. What other connective is logically equivalent to x nor x?
   4. Construct the truth table for:
      (x nor y) nor (x nor y)

   5. What other connective is logically equivalent to that one?

   ---