Let’s tackle the questions step by step! I'll provide detailed solutions and explanations for each.

QUESTION A1

(a) Determine whether the following statements are propositions or not. State their truth values.

(i) "In computers, the largest number you can represent with 8 bits is 11111111."

• Analysis: This is a declarative sentence that can be true or false.
• Proposition: Yes.
• Truth Value: True, as $11111111_2 = 255$ is the largest number in 8 bits.

(ii) "How many rows should a truth table have for a statement involving $n$ different propositions?"

• Analysis: This is a question, not a declarative statement.
• Proposition: No.

(b) Use De Morgan’s Law to write the negation of the statement:

"2 is an even number and prime number."

• Original Statement: $P: \text{2 is even and prime} = (A \land B)$.
• Negation: $\neg P = \neg (A \land B)$.
• Applying De Morgan’s Law:
  $\neg P = \neg A \lor \neg B$, which means
  "2 is not even or 2 is not prime."

QUESTION A2

(a) Determine whether the following statements are true or false:

(i) $2+2=4$ if and only if $3^3 < 27$.

• Analysis: $2+2=4$ is True, but $3^3 = 27$, so $3^3 < 27$ is False.
• Truth Value: False (because for "if and only if," both conditions must hold).

(ii) $\sqrt{4} - 2^1$ is an even number and $1+2 = 4$.

• Analysis:
  • $\sqrt{4} - 2^1 = 2 - 2 = 0$, which is an even number.
  • $1+2 = 3 \neq 4$.
• Truth Value: False.

(iii) $4^3 - 5^2 > 0 \lor (\sqrt{3^4} \leq 20)$.

• Analysis:
  • $4^3 - 5^2 = 64 - 25 = 39 > 0$, so the first part is True.
  • $\sqrt{3^4} = \sqrt{81} = 9 \leq 20$, so the second part is also True.
• Truth Value: True (disjunction requires at least one part to be True).

(b) Rewrite the statement in "if-then" form:

"A sufficient condition for Katrina to take the algorithms course is that she passes discrete mathematics."

• If-then form:
  "If Katrina passes discrete mathematics, then she can take the algorithms course."

QUESTION A3

Show that the Boolean expressions are equal by stating the Boolean laws:
$(x'y + z') + zxy = z + xy + z'$

1. Expand $zxy$ using Distributive Law:
   $(x'y + z') + zxy = (x'y + z') + (zx + zy)$.

2. Combine $z$ terms using Absorption Law:
   $z + z' = 1$ (Complement Law).

3. Simplify further with Idempotent Law:
   $z + z' + xy = z + xy$.

Thus, $(x'y + z') + zxy = z + xy + z'$.

QUESTION A4

(a) Find the output $Q$ of the logic circuit.

• Logic gates:
  • $x \land y \to \neg (\text{output})$: This is the NOT of AND gate.
  • Combine $z$ with results via an OR gate.
  • Final result flows into an AND gate with $z'$.
  • Use truth table or simplify expressions to calculate.

(b) Find the value of $Q$ when $x=0, y=0, z=1$.

1. Substitute the values $x=0, y=0, z=1$ into the circuit.
2. Trace the logic through the gates.

Would you like detailed truth tables, expanded laws, or step-by-step evaluations for any part? Let me know!

5 Follow-Up Questions

1. What is the significance of De Morgan’s Laws in simplifying logical expressions?
2. How do truth tables help in understanding Boolean expressions?
3. What are other examples of statements that are not propositions?
4. How can logic gates be used in real-world applications?
5. What are the practical uses of "if-then" logical reformulations?

Tip

Always double-check Boolean simplifications using truth tables to ensure accuracy!