Finite Math Test 2 Flashcards

The set S for which all of the constraints hold is called the … … or … … [The area you shade on a graph for the solution]
The point in S that optimizes the objective function P is called the … … [The point that is the solution]

The set S for which all of the constraints hold is called the feasible region or feasible set. Each point in S that optimizes the objective function P is called the optimal solution.

If a linear programming problem has a solution, then it must occur at a … or … point of the feasible set S. [Points of the feasible region]

If a linear programming problem has a solution, then it must occur at a vertex or corner point of the feasible set S.

If S is …, then P = ax+by has both a … and a … value on S.

If S is bounded, then P = ax+by has both a max and a min value on S.

If S is … and a >= 0 and b >= 0, then P has a … on S if the constraints include … and …

If S is unbounded and a >= 0 and b >= 0, then P has a min on S if the constraints include x >= 0 and y >= 0

What is the method of corners? [How do you solve a linear programming problem with a graph and equations?]

1. Graph the feasible region S.

2. Evaluate the objective function P at each corner point.

3. The largest such P value is the max (if it exists) and the smallest such P is the min (if it exists).

A … is a well-defined collection of objects. The objects in the … are called … or … of the ….

A set is a well-defined collection of objects. The objects in the set are called elements or members of the set.

Give an example of set-builder notation.

{ x ∈ ℤ | 0 ≤ x ≤ 9 }

Sets V and W are … iff they contain the same …

Sets V and W are equal iff they contain the same elements.

What makes set V a subset of W, and how would you express this symbolically?

Each element in V is also an element of W
V ⊆ W

Describe a proper subset in terms of V and W.

V ⊆ W, but V ≠ W

The … … is the set containing all possible elements of interest in a given context. The … … is the set containing … elements.

The universal set is the set containing all possible elements of interest in a given context. The empty set is the set containing no elements.

By convention, {} is a … of … set.

By convention, {} is a subset of every set.

The … of sets V and W is the set of elements that belong to V … W … both. We write V … W [Is “or“ or “and“ associated with union or intersection?]

The union of sets V and W is the set of elements that belong to V or W or both. We write V U W.

The … of sets V and W is the set of elements that belong to V … W. We write V … W. [Is “or“ or “and” associated with intersection or union?]

The intersection of V and W is the set of elements that belong to V and W. We write V ∩ W.

Sets V and W are … iff V ∩ W = ...

Sets V and W are disjoint iff V ∩ W = {}

The … of a set V is the set of elements in U that … … … to V. We write … [Everything not in V]

The complement of a set V is the set of elements in U that do not belong to V. We write V^C

Describe the 2 DeMorgan’s Laws. [Think in terms of two sets, complements, unions, and intersections…]

1. The complement of the union of two sets is equal to the intersections of the complements of each set.

2. The complement of the intersection of two sets is equal to the union of the complements of each set.

Let A, B be … sets. If A ∩ B = {}, then n(A U B) = … + …

Let A, B be finite sets. If A ∩ B = {}, then n(A U B) = n(A) + n(B)

Let A, B be … sets. Then n(A U B) = … + … - …

Let A, B be finite sets. Then n(A U B) = n(A) + n(B) - n(A ∩ B)

A … of a set of distinct objects is an … of these objects in a definite … [Permutation or combination? and fill in the rest]

A permutation of a set of distinct objects is an arrangement of these objects in a definite order.

The number of permutations of n objects is n… = …!

The number of permutations of n objects is n(n-1)(n-2)…2×1 = n!

The number of permutations (order matters) of … objects taken … at a time is denoted by …

The number of permutations (order matters) of n objects r at a time is denoted by P(n,r).

P(n,r) = …

P(n, r) = n!/(n - r)!

0! = ?

0! = 1

The number of … (order doesn’t matter) of … objects taken … at a time is denoted by … [Permutation or combination? and fill in the rest]

The number of combinations (order doesn’t matter) of n objects taken r at a time is denoted by C(n,r)

C(n,r) = … = …, where r <= n. [Give another way to express C(n,r) and the equation for this]

C(n,r) = (n over r) = n!/r! (n - r)!, where r <= n.