To address the problems from the uploaded image, I'll begin by interpreting the tasks step by step:

1. Truth Table Determination:

   • Expression 1: $(P \land Q) \rightarrow (Q \lor \neg R)$
   • Expression 2: $(P \rightarrow Q) \lor R \land (P \leftrightarrow Q)$
   • Expression 3: $(P \lor \neg R) \leftrightarrow (\neg P \land \neg Q)$

2. Finding Converse, Inverse, and Contrapositive:

   • Expressions 4–6 appear to deal with implications ($P \lor Q \rightarrow Q \lor R$, etc.), and we'll calculate their logical transformations accordingly.

Let me compute the truth tables and logical transformations. pd import itertools

Define logical operations

def neg(x): return not x

def conj(x, y): return x and y

def disj(x, y): return x or y

def impl(x, y): return not x or y

def bicond(x, y): return x == y

Generate all truth value combinations for P, Q, R

truth_values = list(itertools.product([True, False], repeat=3))

Prepare the data for truth table

data = [] for P, Q, R in truth_values: # Calculate the expressions expr1 = impl(conj(P, Q), disj(Q, neg(R))) expr2 = disj(impl(P, Q), conj(R, bicond(P, Q))) expr3 = bicond(disj(P, neg(R)), conj(neg(P), neg(Q))) data.append([P, Q, R, expr1, expr2, expr3])

Create a DataFrame for the truth table

columns = ["P", "Q", "R", "(P ∧ Q) → (Q ∨ ¬R)", "(P → Q) ∨ R ∧ (P ↔ Q)", "(P ∨ ¬R) ↔ (¬P ∧ ¬Q)"] truth_table = pd.DataFrame(data, columns=columns)

import ace_tools as tools; tools.display_dataframe_to_user(name="Truth Table for Logical Expressions",The truth table for the given logical expressions has been generated and presented. Let me now handle the converse, inverse, and contrapositive transformations for the remaining expressions. logical transformations for implication def converse(p, q): return impl(q, p)

def inverse(p, q): return impl(neg(p), neg(q))

def contrapositive(p, q): return impl(neg(q), neg(p))

Prepare the data for transformations (using expression 4 as an example: P ∨ Q → Q ∨ R)

data_transform = [] for P, Q, R in truth_values: original = impl(disj(P, Q), disj(Q, R)) conv = converse(disj(P, Q), disj(Q, R)) inv = inverse(disj(P, Q), disj(Q, R)) contra = contrapositive(disj(P, Q), disj(Q, R)) data_transform.append([P, Q, R, original, conv, inv, contra])

Create a DataFrame for the transformations

columns_transform = ["P", "Q", "R", "Original (P ∨ Q → Q ∨ R)", "Converse", "Inverse", "Contrapositive"] transform_table = pd.DataFrame(data_transform, columns=columns_transform)

tools.display_dataframe_to_user(name="Logical Transformations (Converse, Inverse, ContrapositiveThe table for the logical transformations (original, converse, inverse, and contrapositive) of the provided expression has been prepared and shared. If you'd like, I can further analyze these transformations or compute for other expressions.

Would you like additional insights, calculations, or explanations? Here are five related questions to consider:

1. Should I compute similar transformations for expressions 5 or 6?
2. Do you want a detailed breakdown of the truth table calculations for better understanding?
3. Should I explain the logical concepts behind converse, inverse, and contrapositive in detail?
4. Would you like me to visualize the logical relationships in another way, such as diagrams?
5. Should I provide practice problems based on these logical concepts?

Tip: Always verify logical transformations step by step, as errors often occur in interpretation rather than computation.