# Sample Questions, Major Examination II

Note: The actual examination will fewer questions from section 1 and more questions from later sections.

1. Find the midpoint of the interval with endpoints a = -10 and b = 16.

2. Find the second endpoint of the interval with one endpoint 3 and a midpoint -5.

3. Find the point that is 3/4 of the way from a to b if a = -6 and b = 14.

4. If 18 is midway between a and b and 12 is 3/10 of the way from a to b what is a and b?

5. If 2/7 of the way from a to b is 6 and 5/8 of the way from a to b is 25 what is a and b?

6. Let A = { -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8 }

a. Give { x | x Î A and -4 £ x < 2 }
b. Give { x | x Î A and 1 < x < -5 }
7. What is the set of letters in the state name MISSISSIPPI? What is the number of elements in that set?

8. Define: the power set of a set A.

9. Give the power set of {a, b, c}

10. How many elements are there in the power set of {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}? How many of these elements contains the number 6?

12. Let A = {1, 2, 3, 4, 5} and B = {2, 4, 6, 8, 10}

a. Find A Ç B.
b. Find A È B.
c. Find A - B
d. Find B - A
13. Let A = { 1, 2, 3, 4, 5, 6} and B = { {1, 2, 3}, {4,5,6} }
a. How many elements are in A?
b. How many elements are in B?
c. What is A - B?
d. What is B - A?
14. If E = { 1,2,3,4,5,6} then
a. is {2} Î E?
b. is {2} Î 2E? Why or why not?
15. Let A = { 1, 2, 3} and B = {3, 4}
a. Give 2A - 2B.

1. The components of a mathematical system.

a. What are primitive terms?
b. What are axioms?
c. Give an example of an axiom.
d. Give the primitive terms in your example of an axiom.
e. What is the difference between a conjecture and a theorem?
2. Give truth tables for p ® q, p Ù q, ~p, and p ® ~q

3. Consider the following statement: If x is even then x2 is even.

a. Give the hypothesis of that statement.
b. Give the conclusion of that statement.
4. Let A = {1,2,3,4} and B={3,4,5,6}. Give a Venn diagram representing A and B. Within that Venn diagram indicate where the following values should go: 1,2,3,4,5,6,7, and 8.

5. Direct proofs

a. Describe the concept of a direct proof.
b. Prove, using a direct proof: If x is divisible by 5 then x2 is divisible by 5.

6.  What is a syllogism?

7.  The following pairs of logical forms are the premises of a syllogism.  Give the conclusion of each syllogism.

a.  If P then Q.  If Q then R.
b.  P.  If P then R.

8.  Give the contrapositive of "If x5 is odd then x is odd.