Questions: T T F f f t T F T t t t T F F t t f F T T f f t F T F f f t F F T t f t F F F t f t C. What are the conditions that make the compound statement false? A. No such conditions exist to make the statement false. B. The statement is false when p and r have the same truth values. C. The statement is false when r is true and p and q are false. D. The statement is false when p is true and q and r are false. E. The statement is false when p is false and q and r are true.

T T F f f t
T F T t t t
T F F t t f
F T T f f t
F T F f f t
F F T t f t
F F F t f t
C. What are the conditions that make the compound statement false?
A. No such conditions exist to make the statement false.
B. The statement is false when p and r have the same truth values.
C. The statement is false when r is true and p and q are false.
D. The statement is false when p is true and q and r are false.
E. The statement is false when p is false and q and r are true.

Transcript text: \begin{tabular}{|c|c|c|c|c|c|} \hline$T$ & $T$ & $F$ & $f$ & $f$ & $t$ \\ \hline$T$ & $F$ & $T$ & $t$ & $t$ & $t$ \\ \hline$T$ & $F$ & $F$ & $t$ & $t$ & $f$ \\ \hline$F$ & $T$ & $T$ & $f$ & $f$ & $t$ \\ \hline$F$ & $T$ & $F$ & $f$ & $f$ & $t$ \\ \hline$F$ & $F$ & $T$ & $t$ & $f$ & $t$ \\ \hline$F$ & $F$ & $F$ & $t$ & $f$ & $t$ \\ \hline \end{tabular} C. What are the conditions that make the compound statement false? A. No such conditions exist to make the statement false. B. The statement is false when $p$ and $r$ have the same truth values. C. The statement is false when $r$ is true and $p$ and $q$ are false. D. The statement is false when $p$ is true and $q$ and $r$ are false. E. The statement is false when $p$ is false and $q$ and $r$ are true.

Solution

Step 1: Identify Simple Statements

The simple statements are: p, q, r.

Step 2: Determine Logical Structure

The compound statement is structured as follows: not ((p and q) or (p and r)).

Step 3: Construct Truth Table

{'p': True, 'q': True, 'r': True} => False {'p': True, 'q': True, 'r': False} => False {'p': True, 'q': False, 'r': True} => False {'p': True, 'q': False, 'r': False} => True {'p': False, 'q': True, 'r': True} => True {'p': False, 'q': True, 'r': False} => True {'p': False, 'q': False, 'r': True} => True {'p': False, 'q': False, 'r': False} => True

Step 4: Analyze Conditions

The compound statement is true under the following conditions: {'p': True, 'q': False, 'r': False} {'p': False, 'q': True, 'r': True} {'p': False, 'q': True, 'r': False} {'p': False, 'q': False, 'r': True} {'p': False, 'q': False, 'r': False}

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