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3. (20 points) Propositional Operations and Equivalences All propositional logic statements can be expressed by using \(\neg\), \(\land\), \(\lor\), \(\to\), \(\leftrightarrow\), \(T\) (true) and \(F\) (false). Not all of these 7 symbols are vital to express all statements. For instance instead of \(F\) we can use \(\neg T\), or instead of \(P \land Q\) we can use \(\neg(\neg P \lor \neg Q)\). Actually \(P \leftrightarrow Q\) is usually interpreted as \((P \to Q) \land (Q \to P)\) most of the time. So if we have \(\to\) and \(\land\) we dont need \(\leftrightarrow\) at all. That is to say a shorter list of symbols might be enough to express all possible statements. • Show that \(\to\) and \(F\) is enough to express any propositional logic statement. (Express \(\neg P\), \(P \land Q\), \(P \lor Q\) and \(T\), by using only \(P\), \(Q\), \(\to\) and \(F\).)

          3. (20 points) Propositional Operations and Equivalences
All propositional logic statements can be expressed by using \(\neg\), \(\land\), \(\lor\), \(\to\), \(\leftrightarrow\), \(T\) (true) and \(F\) (false). Not all
of these 7 symbols are vital to express all statements. For instance instead of \(F\) we can use \(\neg T\), or instead
of \(P \land Q\) we can use \(\neg(\neg P \lor \neg Q)\). Actually \(P \leftrightarrow Q\) is usually interpreted as \((P \to Q) \land (Q \to P)\) most of
the time. So if we have \(\to\) and \(\land\) we dont need \(\leftrightarrow\) at all. That is to say a shorter list of symbols might be
enough to express all possible statements.
• Show that \(\to\) and \(F\) is enough to express any propositional logic statement. (Express \(\neg P\), \(P \land Q\), \(P \lor Q\)
and \(T\), by using only \(P\), \(Q\), \(\to\) and \(F\).)

3. (20 points) Propositional Operations and Equivalences
All propositional logic statements can be expressed by using , , , →, ↔, T (true) and F (false). Not all
of these 7 symbols are vital to express all statements. For instance instead of F we can use T, or instead
of P Q we can use ( P Q). Actually P ↔ Q is usually interpreted as (P → Q) (Q → P) most of
the time. So if we have → and we dont need ↔ at all. That is to say a shorter list of symbols might be
enough to express all possible statements.
• Show that → and F is enough to express any propositional logic statement. (Express P, P Q, P Q
and T, by using only P, Q, → and F.)

Added by Melissa Q.

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