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# Cell Phones in the Hands of Drivers: A Risk or a Benefit?

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MLA Research Paper (Levi) Title is centered about one-third down the page. Writer's name is centered around the middle of the page. Course name, professor's name, and date are centered near the bottom of the page. Cell Phones in the Hands of Drivers: A Risk or a Benefit? Paul Levi English 101 Professor Baldwin 2 April XXXX Lopez begins to identify and question Goodall's assumptions. Marginal annotations indicate MLA-style formatting and effective writing.
• scientific reports on the relation between cell phone use
• reckless driving
• cell phones
• drivers
• laws
• use

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Discrete Mathematics
Lecture 5
Harper Langston
New York UniversityEmpty Set
2• S = {x ˛ R, x = -1}
• X = {1, 3}, Y = {2, 4}, C = X ˙ Y
(X and Y are disjoint)
• Empty set has no elements ˘
• Empty set is a subset of any set
• There is exactly one empty set
• Properties of empty set:
– A ¨ ˘ = A, A ˙ ˘ = ˘
c c– A ˙ A = ˘, A ¨ A = U
c c– U = ˘, ˘ = USet Partitioning
• Two sets are called disjoint if they have no
elements in common
• Theorem: A – B and B are disjoint
• A collection of sets A , A , …, A is called 1 2 n
mutually disjoint when any pair of sets from this
collection is disjoint
• A collection of non-empty sets {A , A , …, A } is 1 2 n
called a partition of a set A when the union of
these sets is A and this collection consists of
mutually disjoint setsPower Set
• Power set of A is the set of all subsets of A
• Example on board
• Theorem: if A ˝ B, then P(A) ˝ P(B)
• Theorem: If set X has n elements, then
nP(X) has 2 elements (proof in Section 5.3
– will show if have time)Cartesian Products
• Ordered n-tuple is a set of ordered n
elements. Equality of n-tuples
• Cartesian product of n sets is a set of n-
tuples, where each element in the n-tuple
belongs to the respective set participating
in the productÛ
Û
Ù
Û
Û
Ù
Ú
Û
à
Ù
Ù
Set Properties
• Inclusion of Intersection:
A ˙ B ˝ A and A ˙ B ˝ B
• Inclusion in Union:
A ˝ A ¨ B and B ˝ A ¨ B
• Transitivity of Inclusion:
(A ˝ B B ˝ C) A ˝ C
• Set Definitions:
x ˛ X ¨ Y x ˛ X y ˛ Y
x ˛ X ˙ Y x ˛ X y ˛ Y
x ˛ X – Y x ˛ X y ˇ Y
cx ˛ X x ˇ X
(x, y) ˛ X · Y x ˛ X y ˛ YSet Identities
• Commutative Laws: A ˙ B = A ˙ B and A ¨ B = B ¨ A
• Associative Laws: (A ˙ B) ˙ C = A ˙ (B ˙ C) and (A ¨ B) ¨ C = A ¨ (B ¨
C)
• Distributive Laws:
A ¨ (B ˙ C) = (A ¨ B) ˙ (A ¨ C) and A ˙ (B ¨ C) = (A ˙ B) ¨ (A ˙ C)
• Intersection and Union with universal set: A ˙ U = A and A ¨ U = U
c c• Double Complement Law: (A ) = A
• Idempotent Laws: A ˙ A = A and A ¨ A = A
c c c c c c• De Morgan’s Laws: (A ˙ B) = A ¨ B and (A ¨ B) = A ˙ B
• Absorption Laws: A ¨ (A ˙ B) = A and A ˙ (A ¨ B) = A
c• Alternate Representation for Difference: A – B = A ˙ B
• Intersection and Union with a subset: if A ˝ B, then A ˙ B = A and A ¨ B =
BProving Equality
• First show that one set is a subset of
another (what we did with examples
before)
• To show this, choose an arbitrary
particular element as with direct proofs
(call it x), and show that if x is in A then x
is in B to show that A is a subset of B
• Example (step through all cases)Disproofs, Counterexamples and
Algebraic Proofs
• Is is true that (A – B) ¨ (B – C) = A – C?
(No via counterexample)
• Show that (A ¨ B) – C = (A – C) ¨ (B – C)
(Can do with an algebraic proof, slightly
different)Boolean Algebra
• A Boolean Algebra is a set of elements
together with two operations denoted as +
and * and satisfying the following
properties:
Commutative: a + b = b + a, a * b = b * a
Associative: (a + b) + c = a + (b + c), (a * b) *c = a * (b * c)
Distributive: a + (b * c) = (a + b) * (a + c), a * (b + c) = (a *
b) + (a * c)
Identity: a + 0 = a, a * 1 = a for some distinct unique 0 and
1
Complement: a + ã = 1, a * ã = 0