Solve your doubts and expand your knowledge with IDNLearn.com's extensive Q&A database. Whether your question is simple or complex, our community is here to provide detailed and trustworthy answers quickly and effectively.
Sagot :
Certainly! Let's fill in the truth table step by step to arrive at the final column. We will evaluate each expression for the given combinations of [tex]\( p \)[/tex], [tex]\( q \)[/tex], and [tex]\( r \)[/tex].
#### Step-by-Step Truth Table Analysis:
1. Columns [tex]\( p \leftrightarrow q \)[/tex] and [tex]\( q \leftrightarrow r \)[/tex]:
- The biconditional [tex]\( p \leftrightarrow q \)[/tex] is true if and only if [tex]\( p \)[/tex] and [tex]\( q \)[/tex] have the same truth value.
- Similarly, [tex]\( q \leftrightarrow r \)[/tex] is true if and only if [tex]\( q \)[/tex] and [tex]\( r \)[/tex] have the same truth value.
2. Column [tex]\( p \wedge q \)[/tex]:
- The conjunction [tex]\( p \wedge q \)[/tex] is true if and only if both [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are true.
3. Column [tex]\((p \leftrightarrow q) \wedge (q \leftrightarrow r)\)[/tex]:
- This column is the conjunction of the evaluations of [tex]\( p \leftrightarrow q \)[/tex] and [tex]\( q \leftrightarrow r \)[/tex].
4. Final Column [tex]\((p \leftrightarrow q) \wedge (q \leftrightarrow r) \rightarrow p \wedge q \)[/tex]:
- This column is true if the implication [tex]\((p \leftrightarrow q) \wedge (q \leftrightarrow r) \rightarrow p \wedge q\)[/tex] is true. An implication [tex]\( A \rightarrow B \)[/tex] is true whenever [tex]\( A \)[/tex] is false or [tex]\( B \)[/tex] is true.
Now, let's fill the table using these logical rules.
#### Constructed Truth Table:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|} \hline p & q & r & p \leftrightarrow q & q \leftrightarrow r & p \wedge q & (p \leftrightarrow q) \wedge (q \leftrightarrow r) \rightarrow p \wedge q \\ \hline \text{T} & \text{T} & \text{T} & \text{T} & \text{T} & \text{T} & \text{T} \\ \text{T} & \text{T} & \text{F} & \text{T} & \text{F} & \text{T} & \text{T} \\ \text{T} & \text{F} & \text{T} & \text{F} & \text{F} & \text{F} & \text{T} \\ \text{T} & \text{F} & \text{F} & \text{F} & \text{T} & \text{F} & \text{T} \\ \text{F} & \text{T} & \text{T} & \text{F} & \text{T} & \text{F} & \text{T} \\ \text{F} & \text{T} & \text{F} & \text{F} & \text{F} & \text{F} & \text{T} \\ \text{F} & \text{F} & \text{T} & \text{T} & \text{F} & \text{F} & \text{T} \\ \text{F} & \text{F} & \text{F} & \text{T} & \text{T} & \text{F} & \text{T} \\ \hline \end{array} \][/tex]
This filled truth table gives us a complete analysis of the logical expression [tex]\((p \leftrightarrow q) \wedge (q \leftrightarrow r) \rightarrow p \wedge q\)[/tex] for all possible truth values of [tex]\( p \)[/tex], [tex]\( q \)[/tex], and [tex]\( r \)[/tex].
#### Step-by-Step Truth Table Analysis:
1. Columns [tex]\( p \leftrightarrow q \)[/tex] and [tex]\( q \leftrightarrow r \)[/tex]:
- The biconditional [tex]\( p \leftrightarrow q \)[/tex] is true if and only if [tex]\( p \)[/tex] and [tex]\( q \)[/tex] have the same truth value.
- Similarly, [tex]\( q \leftrightarrow r \)[/tex] is true if and only if [tex]\( q \)[/tex] and [tex]\( r \)[/tex] have the same truth value.
2. Column [tex]\( p \wedge q \)[/tex]:
- The conjunction [tex]\( p \wedge q \)[/tex] is true if and only if both [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are true.
3. Column [tex]\((p \leftrightarrow q) \wedge (q \leftrightarrow r)\)[/tex]:
- This column is the conjunction of the evaluations of [tex]\( p \leftrightarrow q \)[/tex] and [tex]\( q \leftrightarrow r \)[/tex].
4. Final Column [tex]\((p \leftrightarrow q) \wedge (q \leftrightarrow r) \rightarrow p \wedge q \)[/tex]:
- This column is true if the implication [tex]\((p \leftrightarrow q) \wedge (q \leftrightarrow r) \rightarrow p \wedge q\)[/tex] is true. An implication [tex]\( A \rightarrow B \)[/tex] is true whenever [tex]\( A \)[/tex] is false or [tex]\( B \)[/tex] is true.
Now, let's fill the table using these logical rules.
#### Constructed Truth Table:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|} \hline p & q & r & p \leftrightarrow q & q \leftrightarrow r & p \wedge q & (p \leftrightarrow q) \wedge (q \leftrightarrow r) \rightarrow p \wedge q \\ \hline \text{T} & \text{T} & \text{T} & \text{T} & \text{T} & \text{T} & \text{T} \\ \text{T} & \text{T} & \text{F} & \text{T} & \text{F} & \text{T} & \text{T} \\ \text{T} & \text{F} & \text{T} & \text{F} & \text{F} & \text{F} & \text{T} \\ \text{T} & \text{F} & \text{F} & \text{F} & \text{T} & \text{F} & \text{T} \\ \text{F} & \text{T} & \text{T} & \text{F} & \text{T} & \text{F} & \text{T} \\ \text{F} & \text{T} & \text{F} & \text{F} & \text{F} & \text{F} & \text{T} \\ \text{F} & \text{F} & \text{T} & \text{T} & \text{F} & \text{F} & \text{T} \\ \text{F} & \text{F} & \text{F} & \text{T} & \text{T} & \text{F} & \text{T} \\ \hline \end{array} \][/tex]
This filled truth table gives us a complete analysis of the logical expression [tex]\((p \leftrightarrow q) \wedge (q \leftrightarrow r) \rightarrow p \wedge q\)[/tex] for all possible truth values of [tex]\( p \)[/tex], [tex]\( q \)[/tex], and [tex]\( r \)[/tex].
Thank you for being part of this discussion. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Thank you for visiting IDNLearn.com. We’re here to provide dependable answers, so visit us again soon.
