For your upcoming test:

• Know how to verbally and mathematically describe a set
• Know the definition and how to use unions, intersections, and complements
• Know how to represent set(s) with a Venn diagram
  • Know how to shade the appropriate region of a Venn diagram corresponding to a given set
• Know how to use the principle of inclusion-exclusion for counting
• Know how to find the number of objects in a each of the non-overlapping regions of a circle Venn diagram
• Know how and when to use multiplication in counting
• Know the definition of C(n,k), and its formula
• Know Pascal’s Identity: C(n,k) = C(n-1,k) + C(n-1,k-1)
  • C(n,k) counts k-element subsets of the the set {1,2,..,n}. Which of these subsets are counted by C(n-1,k-1) in Pascal’s Identity.
• You should be able to use Pascal’s triangle to compute various C(n,k)
• Know when a counting problem involves counting subsets
  • Examples:
    • Selecting balls and putting them in a box
    • Handshakes
    • Number of matches in a league
    • Many many more!
    • n coin tosses with exactly k heads (see Section 5.6)
    • Routes through a city (see Section 5.6)
• When counting routes in a city (see Section 5.6): Given a route you should be able to write down the subset that corresponds to it.

In general, the best thing to do to study for this test is to review the examples from lecture and homework. The more practice you have, the better you’ll be able to handle the questions on the test!

• Know the steps for solving linear programming problems in two variables

1. Find constraints (this is a list of inequalities) and objective function
2. Graph feasible set. (Suggestion: Find the x and y intercepts of each constraint line), and list the vertices of the feasible set.
3. Test each vertex in the objective function until you find the optimum (minimum or maximum) vertex.

• Be able to carry out each of the steps above
• For applicable linear programming problems: Be able to eliminate all but two variable by using an equation from the word problem (As in the shipping example we did recently.)
• Be able to explain why an interior point in the feasible set is never optimal
• Know how you can find a vertex of the feasible set when it has three variables
  • Be able to solve a linear system, and verify the solution (the intersection point) is feasible or not feasible
• Know the steps involved in the Simplex Method
• Given a recipe for the set of vertices of a feasible set, a recipe for finding the neighbors of a vertex, the objective function and a starting vertex, you should be able to carry out the Simplex Method
• Know the definition of a set and a subset
  • Be able to explain the difference between a set and a list
  • Be able to list all of the subsets of a given set

Make sure that you take time to practice all of the concepts we’ve gone over in class.

The test will focus on solving systems of linear systems, but there will also be questions on graphing linear inequalities.

For true and false questions, if your answer is false I will ask you to give a counter-example. For instance, take the statement: Every system of linear equations has a solution. This is false, and a counter-example is given by the system of equations $latex y=2x +1$ and $latex y=2x +5$. These lines are parallel, and so they never intersect. Thus, this system has no solution.

• Make sure you can graph a feasible set defined by a collection of linear inequalities
• Make sure you know what the elementary row operations.
• Make sure you can change a system of linear equations into an augmented matrix.
• Make sure you know what Gauss-Jordan Elimination means
• If a matrix has infinitely many solutions, make sure you can give at least 2 different solutions to the system using the general solution we compute with Gauss-Jordan Elimination
• Make sure you can perform matrix multiplication
• Make sure you can find the inverse of a matrix using Gauss-Jordan Elimination
• Make sure you can convert a system of linear equations into a matrix equation: $latex AX = B$.
  • Make sure you know how use the inverse to find the solution.
• Make sure you can know that the solution to linear system is the same as the intersection of lines or planes
  • You may want to draw different kinds of intersections we discussed in class: What kind of intersection of planes would give you infinite solutions? What about no solution?

Let me know if you have any questions, and good luck!