Sets and Set Theory: Definition, Representation, Subsets, and Relations - Prof. Dali Wang, Study notes of Discrete Structures and Graph Theory

Download Sets and Set Theory: Definition, Representation, Subsets, and Relations - Prof. Dali Wang and more Study notes Discrete Structures and Graph Theory in PDF only on Docsity!

Logic, Sets

Section 1.6 Sets

Sets^

Set Definition „^ A^ set

is a collection of objects or

elements

or

members

. „^ A set is said to

contain

its elements.

„^ There must be an underlying universal set U, eitherspecifically stated or understood. „ Notation: „^ x is a member of S or x is an element of S:^ x^ ∈^ S „^ x is not an element of S:^ x^ ∉^ S^2 ∈^ {5,-7,

π, “algebra”, 2, 2.718} 8 ∉^ {p : p is a prime number}

Sets^

Common Universal Sets „^ R = reals „^ N = natural numbers = {0,1, 2, 3,... }, the^ counting

numbers „^ Z = all integers = {.. , -3, -2, -1, 0, 1, 2, 3, 4, ...} „^ Z+ is the set of positive integers

Sets^

Subsets & Null Set „^ Definition:

The set A is a

subset

of the set B,

denoted A

⊆^ B, iff

∀x[x^ ∈

A→x^ ∈

B]

„^ Definition:

The^ void

set, the

null^ set, the

empty

set, denoted

∅,^ is the set with no

members.Note: the assertion

x^ ∈ ∅^

is always false. Hence

∀ x [ x^ ∈∅→

x^ ∈ B ] is always true. Therefore,

∅^ is a subset of every

set.Note: Any set is always a subset of itself.

Sets^

More on Sets „^ Definition:

The number of (distinct) elements in A, denoted |A|, is called the

cardinality

of A.

„^ If the cardinality is a natural number (in N),then the set is called

finite , else

infinite

.

„^ Example:A = {a, b},|{a, b}| = 2,|P({a, b})| =|P{1, 3, 5}| =

Sets^

More on Sets „^ Definition:

The^ Cartesian product

of A with B,

denoted A

×B, is the set of ordered pairs {<a, b> |

a

∈ A^ ∧^ b

∈ B } „^ Notation: „^ Example:A = {a,b}, B = {1, 2, 3}AxB = {<a, 1>, <a, 2>, <a, 3>, <b, 1>, <b, 2>, <b, 3>}What is BxA? „^ Example: if |A| = m and |B| = n, what is |AxB|?

}| ,,, {^211

ii n n ii

Aa aa aA X^

∈> <=L=