Questions: Complete the row of each truth table. Use T for true and F for false. p q r r ∧ p q → (r ∧ p) --------------- T T F a b c ~b c → ~b b → a (c → ~b) ∧ (b → a) --------------------- T F T

Solution Steps

To complete the truth tables, we need to evaluate the logical expressions for each row based on the given values of the variables. For the first table, determine the value of \( r \) such that \( r \wedge p = F \), and then evaluate \( q \rightarrow (r \wedge p) \). For the second table, evaluate each logical expression step by step using the given values of \( a \), \( b \), and \( c \).

For the first truth table, we have:

• \( p = T \)
• \( q = T \)
• To satisfy \( r \wedge p = F \), we must have \( r = F \).
• Therefore, \( r \wedge p = F \) results in \( F \).
• Next, we evaluate \( q \rightarrow (r \wedge p) \): \[ q \rightarrow (r \wedge p) = T \rightarrow F = F \]

For the second truth table, we have:

• \( a = T \)
• \( b = F \)
• \( c = T \)
• We calculate \( \sim b \): \[ \sim b = T \]
• Next, we evaluate \( c \rightarrow \sim b \): \[ c \rightarrow \sim b = T \rightarrow T = T \]
• Then, we evaluate \( b \rightarrow a \): \[ b \rightarrow a = F \rightarrow T = T \]
• Finally, we compute \( (c \rightarrow \sim b) \wedge (b \rightarrow a) \): \[ (c \rightarrow \sim b) \wedge (b \rightarrow a) = T \wedge T = T \]

Final Answer