Use the Venn diagram to the right.

a. Write a logical expression for the shaded region.

b. Draw a network where $p, q$, and $r$ are the inputs and the shaded region represents the output.

Use the modern square of opposition to determine whether the following immediate inferences are valid or invalid from the Boolean standpoint. No sculptures by Rodin are boring creations. Therefore, all sculptures by Rodin are boring creations.

How many integers in the set $\{n \in \mathbf{Z} | 1 \leq n \leq 700\}$ are divisible by 2 or 7?

In a class of 40 students, everyone has either a pierced nose or a pierced ear. The professor asks everyone with a pierced nose to raise their hands. Nine hands go up. Then the professor asked everyone with a pierced ear to do likewise. This time there are 34 hands raised. How many students have piercings both on their ears and their noses?

Let A = {1, 3, 5, 7}, let B = {4, 8, 12}, and let the universal set be U = {1, 2, 3, 4, 5, 7, 8, 12, 15}. Is B′ a subset of A?

Let A = {2,3,4}, B = {3,4,5,6}, and suppose the universal set is U = {1,2, ...,9}. List all the elements in the following sets. $(a) (A \cup B)^{\prime}\ (b)(A \cap B) \times A\ { (c) } P(B \backslash A)$

Draw a Venn diagram to show the region $A \cap B^{\prime}.$ This region is also denoted A\B, and is called the set difference, for obvious reasons.

Which of the following is a proprietary protocol developed by Microsoft that provides a user with a graphical interface to another computer?

• A) Secure Sockets Layer (SSL)
• B) Layer 2 Tunneling Protocol (L2TP)
• C) Point-to-Point Tunneling Protocol (PPTP)
• D) Remote Desktop Protocol (RDP)

Rewrite this theorem using two if-then statements:

Two lines are parallel if and only if, when cut by a transversal, corresponding angles have the same measure.

Use the modified Venn diagram technique to determine if the following immediate inference forms are valid from the Boolean standpoint, conditionally valid from the Aristotelian standpoint, or invalid. No A are B. Therefore, some A are not B.

Use the modified Venn diagram technique to determine if the following immediate inference forms are valid from the Boolean standpoint, conditionally valid from the Aristotelian standpoint, or invalid. It is false that some A are not B. Therefore, some A are B.

Write a conditional statement that is logically equivalent to

In order for $2^x>1$ it is sufficient that $x>0$.

Write the negation for the statement given below.

Someone trusts everyone.

Construct the dual graph for the map shown. Then find the number of colors needed to color the map so that no two adjacent regions have the same color.