CS 405G Introduction to
Database Systems
Lecture 2: Discrete Maths for Database
Instructor: Chen Qian
Propositions
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The symbol ∧, called conjunction, and the symbol ∨,
called disjunction, are binary connectives, because each
of them is used to form a compound proposition from
two propositions.
We may call ∧ as “and”, ∨ as “or”.
Truth table:
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Implication
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Implication The binary propositional connective → is
called implication. It represents the combination “if . . .
then.”
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For instance, the assertion “the cube of any positive
number is positive also” can be written as
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Is this formula true or false?
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False, e.g, x=6 is a counter example
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Truth table of implication
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Is this proposition true or false?
“Penguins live in Arctic → KY is a province of Canada”
It is true!
“Penguins live in Arctic → KY is a state of the USA”
Still true!
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Sets
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A set is a collection of objects. We write x ∈ A if object
x is an element of set A, and x A otherwise.
When we specify which objects belong to a set, this
defines the set completely; there is no such thing as the
order of elements in a set or the number of repetitions of
an element in a set. For instance,
If C is a condition, then by {x : C} we denote the set of
all objects x satisfying this condition. For instance,
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Cardinality
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If a set A is finite then the number of elements of A is
also called the cardinality of A and denoted by |A|.
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What is the cardinality of the student set of this class?
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What is the cardinality of the set of exam scores of this
class?
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55
< 55
Cardinality of the natural numbers?
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Subset
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We say that a set A is a subset of a set B, and write A ⊆
B, if every element of A is an element of B. For
instance,
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Operations on Sets
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1. For any sets A and B, by A ∪ B we denote the set
{x : x ∈ A ∨ x ∈ B},
called the union of A and B.
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2. By A ∩ B we denote the set
{x : x ∈ A ∧ x ∈ B},
called the intersection of A and B.
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Operations on Sets
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By A - B we denote the set {x : x ∈ A ∧ x
the difference of A and B.
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B}, called
For instance, {2, 3} - {3, 5} = ?
{2}
The Cartesian product of sets A and B is the set of
ordered pairs <x, y> such that x ∈ A and y ∈ B:
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For instance,
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Power set
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By
we denote the power set of a set A, that is, the
set of all subsets of A:
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For instance
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Relation
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Any condition on a pair of elements of a set A defines a
binary relation, or simply relation, on A.
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A relation R can be characterized by the set of all
ordered pairs <x, y> such that xRy
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For instance, we can say that the relation < on the set {1,
2, 3, 4} is the set
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Relation
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A relation R on a set A is said to be reflexive if, for all
elements x of A, xRx.
We say that R is irreflexive if there is no element x of A
such that xRx.
Are they reflexive or irreflexive?
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=, ≤, <
A relation R on a set A is said to be symmetric if, for all
x, y ∈ A, xRy implies yRx.
Are they symmetric?
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=, ≤, <
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Relation
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A relation R on a set A is said to be transitive if, for all
x, y, z ∈ A, xRy and yRz imply xRz.
Are they transitive?
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=, <, ≤,
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Exercise
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Consider the relation x = 2y + 1 between real numbers
x, y. Is it reflexive? Is it symmetric? Is it transitive?
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This relation is not reflexive, because the condition 1R1
does not hold.
This relation is not symmetric, because the condition
1R3 holds, but the condition 3R1 doesn’t.
This relation is not transitive, because the conditions
1R3 and 3R7 hold, but the condition 1R7 doesn’t.
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Summary
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Propositions
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Sets
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Operations on sets
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Relations
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Next class: ER model
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Credit
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Some slides are made from the lecture notes of Prof.
Vladimir Lifschitz.
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