Q11-12 points Consider the poset (D​, | ), where D​ ={1, 2, 3, 5, 6, 10, 15, 30}.                                                                                                                    (Note: “ | ” is the symbol for “is divisible by”.) ​                                            (a) Find all lower bounds of 6 and 10 taken collectively. Do not find the lower bounds of 6 and 10 separately.                                                           (b) Find the greatest lower bound of 6 and 10 taken collectively. Do not find the greatest lower bound of 6 and 10 separately.                                                        (c) Determine the least upper bound of 2 and 3 taken collectively. Do not find the least upper bound of 2 and 3 separately.                                                        (d) Draw the Hasse diagram for this poset. ​                                                   (e) Determine the complement of each element of D in [D; ⋁ , ⋀ ]. (f) Is the lattice for [D​; ⋁ , ⋀ ] a Boolean algebra? If so, why?

Q7-10 points Consider the function  f : ℝ → ℝ defined by f (x) = x³ – 2x + 3. Prove or provide a counter example that this function is onto (or surjective). Prove or provide a counter example that this function is one-to-one (or injective). 

Q11-5 points The standard telephone number in the U.S. has 10 digits; for example, NOVA’s telephone number is (703) 323-3000. How many phone numbers are possible for the 703 area code if the first digit that follows the 703 cannot be a 0 or a 1?

Q3-10 points Let ​A​ = {1, 2, 3, 4} and let ​R​ be the relation with digraph Test this relation for the following properties: reflexive symmetric transitive

Q3-5 points Given the following sets: U = ​{1, 2, 3, 4, 5, 6, 7, 8} A = ​{4, 6, 7} B = ​{1, 2, 4, 7} C = ​{1, 3, 5, 6, 8} Let D = C′    (the complement of set C, sometimes denoted CC  or C.) Find the power set of D, P(D).

Q1-8 points Let ​A​ = {1, 2, 3, 4} and let ​R​ be the following relation from ​A ​to​ A​: R​ = {(1, 1), (1, 4), (2, 1), (2, 2), (2, 3), (3, 1), (3, 4), (4, 3)} ​                                                                                                                        Create the digraph for this relation. Create the matrix ​MR​ ​for this relation. ​

Q16-5 points Use the Binomial Theorem to write out the expansion of (2a + b)3 .

Q15-5 points Make a Venn Diagram that illustrates the set (​A​ ​∪​ B​) ∩ C.

Q1-5 points Given the following sets: U = ​{1, 2, 3, 4, 5, 6, 7, 8} A = ​{4, 6, 7} B = ​{1, 2, 4, 7} C = ​{1, 3, 5, 6, 8} Find C U B.

 Q13-5 points Jim has a drawer with 6 red socks, 6 blue socks, and 8 black socks. If he picks them out without looking, how many socks must Jim pull out of the drawer to guarantee that he has a matching color pair?