Answer:
Certainly! Below are the truth tables for the given sentence-types:
Conjunction (p • q):
p q p • q
T T T
T F F
F T F
F F F
Disjunction (p ∨ q):
p q p ∨ q
T T T
T F T
F T T
F F F
Conditional (if p then q):
p q p → q
T T T
T F F
F T T
F F T
Biconditional (p ↔ q):
p q p ↔ q
T T T
T F F
F T F
F F T
These truth tables illustrate the possible combinations of truth values for the propositions p and q, as well as the resulting truth values for each logical operation.
Answer:
See below
Step-by-step explanation:
To create truth tables for the given sentence types, we will evaluate them for all possible combinations of truth values of the propositional variables [tex]\sf p [/tex] and [tex]\sf q [/tex].
Conjunction (p • q):
The conjunction [tex]\sf p \land q [/tex] is true if and only if both [tex]\sf p [/tex] and [tex]\sf q [/tex] are true; otherwise, it is false.
[tex]\begin{aligned}\begin{array}{|c|c|c|}\hline p & q & p \land q \\\hline\text{T} & \text{T} & \text{T} \\\text{T} & \text{F} & \text{F} \\\text{F} & \text{T} & \text{F} \\\text{F} & \text{F} & \text{F} \\\hline\end{array}\end{aligned}[/tex]
Disjunction (p ∨ q):
The disjunction [tex]\sf p \lor q [/tex] is false if and only if both [tex]\sf p [/tex] and [tex]\sf q [/tex] are false; otherwise, it is true.
[tex] \begin{array}{|c|c|c|}\hline p & q & p \lor q \\\hline\text{T} & \text{T} & \text{T} \\\text{T} & \text{F} & \text{T} \\\text{F} & \text{T} & \text{T} \\\text{F} & \text{F} & \text{F} \\\hline\end{array}[/tex]
Implication (if p then q):
The implication [tex]\sf p \rightarrow q [/tex] is false if [tex]\sf p [/tex] is true and [tex]\sf q [/tex] is false; otherwise, it is true.
[tex] \begin{array}{|c|c|c|}\hline p & q & p \rightarrow q \\\hline\text{T} & \text{T} & \text{T} \\\text{T} & \text{F} &\text{F} \\\text{F} & \text{T} & \text{T} \\\text{F} & \text{F} & \text{T} \\\hline\end{array} [/tex]
Biconditional (p = q):
The biconditional [tex]\sf p \leftrightarrow q [/tex] is true if [tex]\sf p [/tex] and [tex]\sf q [/tex] have the same truth value; otherwise, it is false.
[tex] \begin{array}{|c|c|c|}\hline p & q & p \leftrightarrow q \\\hline\text{T} & \text{T} & \text{T} \\\text{T} & \text{F} & \text{F} \\\text{F} & \text{T} & \text{F} \\\text{F} & \text{F} & \text{T} \\\hline\end{array} [/tex]
In each table:
[tex]\sf \textsf{T} [/tex] represents "True".
[tex]\sf \textsf{F} [/tex] represents "False".
[tex]\sf p [/tex] and [tex]\sf q [/tex] are the truth values of the propositional variables [tex]\sf p [/tex] and [tex]\sf q [/tex], respectively.
The resulting column shows the truth value of the given logical expression based on the truth values of [tex]\sf p [/tex] and [tex]\sf q [/tex].
