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solution. Task 1. Truth Tables and Logical Equivalencies [10 points] If p and q are propositions, then we can define $p \oplus q$ as the statement $(p \land \neg q) \lor (\neg p \land q)$. In other words, $p \oplus q$ is true if and only if p is true and q is false, or p is false and q is true. The connective $\oplus$ stands for Exclusive OR or XOR. 1. Write the truth table of $\oplus$, resorting to its definition provided above. Indicate how you derive the resulting $\oplus$-column by additional columns. 2. Is $\oplus$ an idempotent connective? In other words, is the following logical equivalence $p \oplus p = p$ true? 3. Prove that $\oplus$ is an associative connective, in other words show that $(p \oplus q) \oplus r = p \oplus (q \oplus r)$. 4. Use logical equivalence rules to show that $\neg p \lor (p \land \neg q) \to q = (p \land q) \lor q$.

          solution.
Task 1. Truth Tables and Logical Equivalencies [10 points]
If p and q are propositions, then we can define $p \oplus q$ as the statement $(p \land \neg q) \lor (\neg p \land q)$. In
other words, $p \oplus q$ is true if and only if p is true and q is false, or p is false and q is true. The
connective $\oplus$ stands for Exclusive OR or XOR.
1. Write the truth table of $\oplus$, resorting to its definition provided above. Indicate how you
derive the resulting $\oplus$-column by additional columns.
2. Is $\oplus$ an idempotent connective? In other words, is the following logical equivalence
$p \oplus p = p$ true?
3. Prove that $\oplus$ is an associative connective, in other words show that $(p \oplus q) \oplus r = p \oplus (q \oplus r)$.
4. Use logical equivalence rules to show that $\neg p \lor (p \land \neg q) \to q = (p \land q) \lor q$.

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solution.
Task 1. Truth Tables and Logical Equivalencies [10 points]
If p and q are propositions, then we can define p ⊕ q as the statement (p q) ( p q). In
other words, p ⊕ q is true if and only if p is true and q is false, or p is false and q is true. The
connective ⊕ stands for Exclusive OR or XOR.
1. Write the truth table of ⊕, resorting to its definition provided above. Indicate how you
derive the resulting ⊕-column by additional columns.
2. Is ⊕ an idempotent connective? In other words, is the following logical equivalence
p ⊕ p = p true?
3. Prove that ⊕ is an associative connective, in other words show that (p ⊕ q) ⊕ r = p ⊕ (q ⊕ r).
4. Use logical equivalence rules to show that p (p q) → q = (p q) q.

Added by Shawn R.

Computer Science and Information Technology

Trishna Knowledge Systems 2018 Edition

Instant Answer

Solved by Expert Willis James

02/22/2021

Step 1

Step 1: We can construct the truth table for $\oplus$ by first constructing columns for $p$, $q$, $\neg p$, $\neg q$, $p \land \neg q$, $\neg p \land q$, and finally $p \oplus q$. Show more…

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solution. Task 1. Truth Tables and Logical Equivalencies [10 points] If p and q are propositions, then we can define $p \oplus q$ as the statement $(p \land \neg q) \lor (\neg p \land q)$. In other words, $p \oplus q$ is true if and only if p is true and q is false, or p is false and q is true. The connective $\oplus$ stands for Exclusive OR or XOR. 1. Write the truth table of $\oplus$, resorting to its definition provided above. Indicate how you derive the resulting $\oplus$-column by additional columns. 2. Is $\oplus$ an idempotent connective? In other words, is the following logical equivalence $p \oplus p = p$ true? 3. Prove that $\oplus$ is an associative connective, in other words show that $(p \oplus q) \oplus r = p \oplus (q \oplus r)$. 4. Use logical equivalence rules to show that $\neg p \lor (p \land \neg q) \rightarrow q = (p \land q) \lor q$.

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