Basic Structures - Discrete Mathematics - Lecture Slides, Slides of Discrete Mathematics

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CSE115/ENGR160 Discrete Mathematics 02/09/

2. 1 Basic structures

• Sets
• Functions
• Sequences
• Sums

Notation

• The set of all vowels in the English alphabet can be written as V={a, e, i, o, u}
• The set of odd positive integers less than 10 can be expressed by O={1, 3, 5, 7, 9}
• Nothing prevents a set from having seemingly unrelated elements, {a, 2, Fred, New Jersey}
• The set of positive integers<100: {1,2,3,…, 99}

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a ∈ A : a isan elemnetof theset A. a ∉ A : otherwise

Notation

• Set builder : characterize the elements by stating the property or properties
• The set O of all odd positive integers < 10:

O={x|x is an odd positive integer < 10} or specify as

• The set of positive rational numbers

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O = { x ∈ Z +^ | x isoddand x < 10 }

Q + ={ x ∈ R | x = p / q forsomepositiveintegers p and q }

Sets and operations

• A datatype or type is the name of a set,
• Together with a set of operations that can be performed on objects from that set
• Boolean : the name of the set {0,1} together with operations on one or more elements of this set such as AND, OR, and NOT

Sets

• Two sets are equal if and only if they have the same elements
• That is if A and B are sets, then A and B are equal if and only if. We write A=B if A and B are equal sets
• The sets {1, 3, 5} and {3, 5, 1} are equal
• The sets {1, 3, 3, 3, 5, 5, 5, 5} is the same as {1, 3, 5} because the have the same elements

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∀ x ( x ∈ A ↔ x ∈ B )

Empty set and singleton

• Empty ( null ) set: denoted by {} or Ø
• The set of positive integers that are greater than their squares is the null set
• Singleton : A set with one element
• A common mistake is to confuse Ø with {Ø}

Subset

• The set A is a subset of B if and only if every element of A is also an element of B
• Denote by A⊆B
• We see A⊆B if and only if

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∀ x ( x ∈ A → x ∈ B )

Proper subset

• A is a proper subset of B: Emphasize that A is a subset of B but that A≠B, and write it as A⊂B
• One way to show that two sets have the same elements is to show that each set is a subset of the other, i.e., if A⊆B and B⊆A, then A=B

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∀ x ( x ∈ A → x ∈ B )∧∃ x ( x ∈ B ∧ x ∉ A )

∀ x ( x ∈ A ↔ x ∈ B )

Sets have other sets as members

• A={∅, {a}, {b}, {a,b}}
• B={x|x is a subset of the set {a, b}}
• Note that A=B and {a}∊A but a∉A
• Sets are used extensively in computing problem

Infinite set and power set

• A set is said to be infinite if it is not finite
  • The set of positive integers is infinite
• Given a set S, the power set of S is the set of all subsets of the set S. The power set of S is denoted by P(S)
• The power set of {0,1,2}
  • P({0,1,2})={∅,{0},{1},{2},{0,1},{1,2},{0,2},{0,1,2}}
  • Note the empty set and set itself are members of this set of subsets

Example

• What is the power set of the empty set?
  • P(∅)={∅}
• The set {∅} has exactly two subsets, i.e., ∅, and the set {∅}. Thus P({∅})={∅,{∅}}
• If a set has n elements, then its power set has 2n^ elements

Ordered pairs

• 2-tupels are called ordered pairs
• (a, b) and (c, d) are equal if and only if a=c and b=d
• Note that (a, b) and (b, a) are not equal unless a=b

Cartesian product

• The Cartesian product of sets A and B, denoted by A x B, is the set of all ordered pairs (a,b), where a ∊ A and b ∊ B
• A: students of UC Merced, B: all courses offered at UC Merced
• A x B consists of all ordered pairs of (a, b), i.e., all possible enrollments of students at UC Merced 20

A × B ={( a , b )| a ∈ A ∧ b ∈ B }