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5:05 B () \( \mathbf{A} \) \( 95 \% \) Practice Exercise 4 I. By constructing truth tables, determine if the given proposition is a tautology, contradiction, or contingency. a. \( (p \wedge q) \rightarrow p \) b. \( -(p \rightarrow q) \leftrightarrow \neg p \vee q \) c. \( \lfloor\neg p \wedge(p \vee q)\rfloor \vee \neg q \) d. \( [(p \rightarrow q) \wedge(q \rightarrow r)] \rightarrow(p \rightarrow r) \) e. \( [p \vee \neg(q \wedge r)] \leftrightarrow((p \wedge \neg r) \wedge \neg q) \) Practice Exercise 4 II. By constructing truth tables, prove the each of the following logical implications. a. \( (p \wedge q) \Rightarrow(p \leftrightarrow q) \) b. \( q \Rightarrow\lfloor(p \wedge q) \leftrightarrow p\rfloor \) c. \( \lfloor(p \vee q) \wedge(p \rightarrow r) \wedge(q \rightarrow r)\rfloor \Rightarrow r \) III. By constructing truth tables, prove the each of the following logical equivalences. a. \( -(p \leftrightarrow q) \equiv p \leftrightarrow \neg q \) b. \( \neg p \rightarrow(q \rightarrow r) \equiv q \rightarrow(p \vee r) \) Practice Exercise 4 IV. Write the rule of replacement in the following proof of logical equivalences. \[ (p \rightarrow q) \wedge(p \rightarrow r) \equiv p \rightarrow(q \cap r) \] Proof: \( (p \rightarrow q) \wedge(p \rightarrow r) \equiv(\neg p \vee q) \wedge(\neg p \vee r) \) \[ \begin{array}{l} \equiv \neg p \vee(q \wedge r) \\ \equiv p \rightarrow(q \wedge r) \end{array} \] V. Prove the following logical equivalences using rules of replacement. a. \( (p \rightarrow r) \vee(q \rightarrow r) \equiv(p \wedge q) \rightarrow r \) b. \( \lfloor(p \wedge q) \rightarrow r\rfloor \equiv\lfloor(p \wedge \neg r) \rightarrow \neg q\rfloor \) References Rosen, Kenneth H. (2008). Discrete Mathematics and its Applications. \( 6^{\text {th }} \) edition . Boston: Mc-Graw Hill. Dacanay, Elvena, Lupague (2011). Introduction to Mathematical Logic. Manila: Mc-Graw Hill.

          5:05 B () \( \mathbf{A} \)
\( 95 \% \)
Practice Exercise 4
I. By constructing truth tables, determine if the given proposition is a tautology, contradiction, or contingency.
a. \( (p \wedge q) \rightarrow p \)
b. \( -(p \rightarrow q) \leftrightarrow \neg p \vee q \)
c. \( \lfloor\neg p \wedge(p \vee q)\rfloor \vee \neg q \)
d. \( [(p \rightarrow q) \wedge(q \rightarrow r)] \rightarrow(p \rightarrow r) \)
e. \( [p \vee \neg(q \wedge r)] \leftrightarrow((p \wedge \neg r) \wedge \neg q) \)
Practice Exercise 4
II. By constructing truth tables, prove the each of the following logical implications.
a. \( (p \wedge q) \Rightarrow(p \leftrightarrow q) \)
b. \( q \Rightarrow\lfloor(p \wedge q) \leftrightarrow p\rfloor \)
c. \( \lfloor(p \vee q) \wedge(p \rightarrow r) \wedge(q \rightarrow r)\rfloor \Rightarrow r \)
III. By constructing truth tables, prove the each of the following logical equivalences.
a. \( -(p \leftrightarrow q) \equiv p \leftrightarrow \neg q \)
b. \( \neg p \rightarrow(q \rightarrow r) \equiv q \rightarrow(p \vee r) \)
Practice Exercise 4
IV. Write the rule of replacement in the following proof of logical equivalences.
\[
(p \rightarrow q) \wedge(p \rightarrow r) \equiv p \rightarrow(q \cap r)
\]

Proof: \( (p \rightarrow q) \wedge(p \rightarrow r) \equiv(\neg p \vee q) \wedge(\neg p \vee r) \)
\[
\begin{array}{l}
\equiv \neg p \vee(q \wedge r) \\
\equiv p \rightarrow(q \wedge r)
\end{array}
\]
V. Prove the following logical equivalences using rules of replacement.
a. \( (p \rightarrow r) \vee(q \rightarrow r) \equiv(p \wedge q) \rightarrow r \)
b. \( \lfloor(p \wedge q) \rightarrow r\rfloor \equiv\lfloor(p \wedge \neg r) \rightarrow \neg q\rfloor \)
References
Rosen, Kenneth H. (2008). Discrete Mathematics and its Applications. \( 6^{\text {th }} \) edition . Boston: Mc-Graw Hill.
Dacanay, Elvena, Lupague (2011). Introduction to Mathematical Logic. Manila: Mc-Graw Hill.

5:05 B () 𝐀
95 %
Practice Exercise 4
I. By constructing truth tables, determine if the given proposition is a tautology, contradiction, or contingency.
a. (p ∧ q) → p
b. -(p → q) ↔ p ∨ q
c. ⌊ p ∧(p ∨ q)⌋∨ q
d. [(p → q) ∧(q → r)] →(p → r)
e. [p ∨(q ∧ r)] ↔((p ∧ r) ∧ q)
Practice Exercise 4
II. By constructing truth tables, prove the each of the following logical implications.
a. (p ∧ q) ⇒(p ↔ q)
b. q ⇒⌊(p ∧ q) ↔ p⌋
c. ⌊(p ∨ q) ∧(p → r) ∧(q → r)⌋⇒ r
III. By constructing truth tables, prove the each of the following logical equivalences.
a. -(p ↔ q) ≡ p ↔ q
b. p →(q → r) ≡ q →(p ∨ r)
Practice Exercise 4
IV. Write the rule of replacement in the following proof of logical equivalences.

(p → q) ∧(p → r) ≡ p →(q ∩ r)

Proof: (p → q) ∧(p → r) ≡( p ∨ q) ∧( p ∨ r)

≡ p ∨(q ∧ r)
≡ p →(q ∧ r)

V. Prove the following logical equivalences using rules of replacement.
a. (p → r) ∨(q → r) ≡(p ∧ q) → r
b. ⌊(p ∧ q) → r⌋≡⌊(p ∧ r) → q⌋
References
Rosen, Kenneth H. (2008). Discrete Mathematics and its Applications. 6^th edition . Boston: Mc-Graw Hill.
Dacanay, Elvena, Lupague (2011). Introduction to Mathematical Logic. Manila: Mc-Graw Hill.

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