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MATH 240; FINAL EXAM, 150 points, 20 December, 2005 (R.A.Brualdi)
TOTAL SCORE (10 problems; plus one 15 point Bonus):
Name:
TA: Anders Hendrickson (circle time) Mon 12:05 Mon 1:20 Wed 12:05 Wed. 1:
Do NOT compute factorials or binomial coefficients.
- [10 points] Let A be a set of 7 elements and let B be a set of 9 elements. What are:
- The number of injective (one-to-one) functions f from A to B:
- The number of binary relations R from A to B:
- [10 points] Let A = {a, b, c}. Draw the diagram of the partially ordered (S, ⊆) where S is the power set of A. Then topologically sort the elements of this partially ordered set.
- [15 points] Twenty-four people are to be transported by 6 cars (a Toyota, a Subaru, a Ford, a Jeep, a Chevrolet, and a Chrysler) with 4 people per car.
- In how many ways can the transportation be arranged?
- If one person in each car is to have a designated driver, how many ways can the transportation be arranged?
- [10 points] For n ≥ 1, Let hn denote the number of ways for a person to climb a flight of n stairs when the person takes 1, 3, or 4 steps at a time. What is a recurrence relation for hn with initial conditions?
- [30 points] Consider the binary relations R on a set A defined below.
- What are the three defining properties for R to be an equivalence relation?
- What are the three defining properties for R to be a partial order?
- Let R be the relation on A = {− 3 , − 2 , − 1 , 0 , 1 , 2 , 3 } defined by aRb provided |a| = |b| (absolute value). Is R an equivalence relation, total order, partial order, or none of these? (circle one) If an equivalence relation, how many different equivalence classes are there and what are they?
- Let R be the relation on A = { 1 , 2 , 3 ,... , 20 } defined by aRb provided that a is a divisor of b. Is R an
equivalence relation, total order, partial order, or none of these? (circle one). If an equivalence relation, how many different equivalence classes are there and what are they?
- Let R be the relation on the set A = { 1 , 2 , 3 ,... , 19 , 20 } of 20 elements defined by aRb provided that a ≡ b (mod) 6. Is R an
equivalence relation, total order, partial order, or none of these? (circle one)
If an equivalence relation, how many different equivalence classes are there and what are they?
- [10 points] Let A be a set of 4 elements, and let R be a binary relation on A.
- If R is a symmetric relation and MR has 0’s and 1’s as shown:
, then
show what the other entries of MR are.
- Let MR have 0’s and 1’s in positions as shown:
. The number of ways
to complete this to an anti-symmetric, binary relation on A equals:
- [20 points] Consider the poset (S, ≤) whose diagram is given below.
Determine
- All maximal elements.
- All minimal elements.
- All the upper bounds of a and b, and then the LUB of a and b if it exists.
- All the lower bounds of a and b, and then the GLB of a and b if it exists.
- [20 points]
- The following is the postfix (postorder) form of a logical expression. What is its (unambiguous) usual form?
p q r ∨ ∧ q p∧ → p r ∧ ∨
- What is the prefix (preorder) form of the algebraic expression
((x × y) + z) − ((z × (u + v)) + ((y × z) × (x + y)))
- Prove by structural induction that n(T ) ≤ 2 h(T^ )+1^ − 1 a FBT T.
