The statement pattern $(p \wedge q) \wedge [(p \wedge q) \vee (\sim p \wedge q)]$ is equivalent to

The logically equivalent statement of $(\sim p \wedge q) \vee (\sim p \wedge \sim q) \vee (p \wedge \sim q)$ is

The dual of the statement $[p \vee (\sim q)] \wedge (\sim p)$ is

Which of the following statement patterns is a tautology?

Show that the statement "For any real numbers $a$ and $b$,$a^{2}=b^{2}$ implies that $a=b$" is not true by giving a counter-example.

Write the following statement in the form of "if-then":
To get an $A^{+}$ in the class,it is necessary that you do all the exercises of the book.