Assignment Brief: Find the truth value of ((¬ P /\ Q) ⇒ (R ⇔ ¬ Q)) \/ S if P is true, Q is true, R is false, and S is true. What is the truth value for the expression if the brackets are omitted? Rewrite the statement removing any unnecessary brackets and explain which (if any) brackets can be omitted or have to be kept.

Assignment Brief:

1. Find the truth value of ((¬ P /\ Q) ⇒ (R ⇔ ¬ Q)) \/ S if P is true, Q is true, R is false, and S is true. What is the truth value for the expression if the brackets are omitted? Rewrite the statement removing any unnecessary brackets and explain which (if any) brackets can be omitted or have to be kept.

2. Consider the proposition:

S/\(S⇒B)/\(¬S⇒¬T)⇒T\/B

• Translate the proposition into natural English by interpreting the propositional variables as the atomic sentences:

S Today is sunny. B I go to the beach. T  I am tired.

• Construct a truth table and check whether the argument is valid.

3. Consider the following collection of statements:

The loop is working or not working. If the loop is working then the program will compile. If the program can compile then the error has been resolved. Therefore, if the error has not been resolved, then the loop is not working.

• Identify the atomic statements and choose appropriate symbols for them. Identify the hypotheses and the conclusion and express them in propositional logic notation.

• Construct a formal proof using a table containing assertions and justifications showing that from the hypotheses follows the conclusion. Explain if you did not use a direct proof.

4. Prove by induction that the sum of the first n odd numbers is equal to the n^th square.
5. Prove by induction that 23  − 1 is divisible by 7, where n is a natural number.
6. Construct a formal proof to show that: A⇒B\/C, ¬A⇒D,¬C,¬D⊢B.