Discrete Mathematics - Harvard Mathematics Department
... Functions Definition If f is a function from A to B, we say that A is its domain and B its codomain. If f (a) = b, we say that b is the image of a and a is the preimage of b. The range or image of f is the set of all the images of all the elements of A. Also, we sometimes say f maps A to B. ...
... Functions Definition If f is a function from A to B, we say that A is its domain and B its codomain. If f (a) = b, we say that b is the image of a and a is the preimage of b. The range or image of f is the set of all the images of all the elements of A. Also, we sometimes say f maps A to B. ...
Notes on Discrete Mathematics
... 3.5.3 Functions of more (or less) than one argument 3.5.4 Composition of functions . . . . . . . . . . . . 3.5.5 Functions with special properties . . . . . . . . 3.5.5.1 Surjections . . . . . . . . . . . . . . . 3.5.5.2 Injections . . . . . . . . . . . . . . . . 3.5.5.3 Bijections . . . . . . . . . ...
... 3.5.3 Functions of more (or less) than one argument 3.5.4 Composition of functions . . . . . . . . . . . . 3.5.5 Functions with special properties . . . . . . . . 3.5.5.1 Surjections . . . . . . . . . . . . . . . 3.5.5.2 Injections . . . . . . . . . . . . . . . . 3.5.5.3 Bijections . . . . . . . . . ...
Discrete Mathematics
... 8.12) A U.S. SSN is a 9-digit number. The first digit(s) may be 0. The numbers are in groups AAA − BB − CCCC. No group can be all zeroes. a) How many SSNs are available? If nothing was un-allowed, there would be 109 . Then we can figure out how many of each un-allowed case there are. 8.7) I have 30 ...
... 8.12) A U.S. SSN is a 9-digit number. The first digit(s) may be 0. The numbers are in groups AAA − BB − CCCC. No group can be all zeroes. a) How many SSNs are available? If nothing was un-allowed, there would be 109 . Then we can figure out how many of each un-allowed case there are. 8.7) I have 30 ...
Document
... Divide-and-Conquer Algorithms and Recurrence Relations • Example 4: Fast Multiplication of Integers • There are more efficient algorithms than the conventional algorithm (described in section 3.6) for multiplying integers. • Suppose that a and b are integers with binary expansions of length 2n. • L ...
... Divide-and-Conquer Algorithms and Recurrence Relations • Example 4: Fast Multiplication of Integers • There are more efficient algorithms than the conventional algorithm (described in section 3.6) for multiplying integers. • Suppose that a and b are integers with binary expansions of length 2n. • L ...
Finite and Infinite Sets
... Definition. A set A is a finite set provided that A D ; or there exists a natural number k such that A Nk . A set is an infinite set provided that it is not a finite set. If A Nk , we say that the set A has cardinality k (or cardinal number k), and we write card .A/ D k. In addition, we say that ...
... Definition. A set A is a finite set provided that A D ; or there exists a natural number k such that A Nk . A set is an infinite set provided that it is not a finite set. If A Nk , we say that the set A has cardinality k (or cardinal number k), and we write card .A/ D k. In addition, we say that ...
Logic and Mathematical Reasoning
... Does it surprise you that we only have two quantifiers? At first it might seem strange but what are the other possibilities? In general, it is not very useful to embed the idea “there are 5 x satisfying P (x),” directly into our logic. This is for a variety of reasons including • we don’t want to ha ...
... Does it surprise you that we only have two quantifiers? At first it might seem strange but what are the other possibilities? In general, it is not very useful to embed the idea “there are 5 x satisfying P (x),” directly into our logic. This is for a variety of reasons including • we don’t want to ha ...
Real Analysis - user web page
... Example . The Cartesian product of the set of real numbers with it self gives the set of all order pairs of real numbers. We call this set the plane. Definition. Let A and B be any two sets. A function f from A into B is a subset of AXB with the property that each x ∈ A is the firs component of prec ...
... Example . The Cartesian product of the set of real numbers with it self gives the set of all order pairs of real numbers. We call this set the plane. Definition. Let A and B be any two sets. A function f from A into B is a subset of AXB with the property that each x ∈ A is the firs component of prec ...
Document
... sets. Really, cardinality is a much deeper concept. Cardinality allows us to generalize the notion of number to infinite collections and it turns out that many type of infinities exist. EG: ...
... sets. Really, cardinality is a much deeper concept. Cardinality allows us to generalize the notion of number to infinite collections and it turns out that many type of infinities exist. EG: ...
Equivalence Relations
... The concept of equivalence relation is an abstraction of the idea of two math objects being like each other in some respect. If an object a is like an object b in some specified way, then b is like a in that respect. a is like itself in every respect! So if you want to give an abstract definitio ...
... The concept of equivalence relation is an abstraction of the idea of two math objects being like each other in some respect. If an object a is like an object b in some specified way, then b is like a in that respect. a is like itself in every respect! So if you want to give an abstract definitio ...
The least known
... Is it possible to use less number of transpositions for obtaining all n! permutations? Is it possible to fix the sequence of transpositions by the only way for all products? ...
... Is it possible to use less number of transpositions for obtaining all n! permutations? Is it possible to fix the sequence of transpositions by the only way for all products? ...
Notes on Combinatorics - School of Mathematical Sciences
... Subsets and binomial coefficients One of the features of combinatorics is that there are usually several different ways to prove something: typically, by a counting argument, or by analytic methods. There are lots of examples below. If two proofs are given, study them both. Combinatorics is about te ...
... Subsets and binomial coefficients One of the features of combinatorics is that there are usually several different ways to prove something: typically, by a counting argument, or by analytic methods. There are lots of examples below. If two proofs are given, study them both. Combinatorics is about te ...
Slides for Rosen, 5th edition - Homepages | The University of
... • Intuitively, this is the function that undoes everything that f does • Formally, it’s the unique function such that f 1 f I A (the identity function on A) ...
... • Intuitively, this is the function that undoes everything that f does • Formally, it’s the unique function such that f 1 f I A (the identity function on A) ...
CS1231 - Lecture 09
... Which is more infinite? Most would answer intuitively that Z contains more elements than Z+. But your intuition must be in subjection to logic. And logic tells you that they both contain the same amount of elements: |Z|;|Z+| (We can also write: |Z| = |Z+| ) ...
... Which is more infinite? Most would answer intuitively that Z contains more elements than Z+. But your intuition must be in subjection to logic. And logic tells you that they both contain the same amount of elements: |Z|;|Z+| (We can also write: |Z| = |Z+| ) ...
Document
... Consider the poset (S,≤), where S = {2, 4, 5, 10, 15, 20} and the partial order ≤ is the divisibility relation In this poset, there is no element b ∈ S such that b 5 and b divides 5. (That is, 5 is not divisible by any other element of S except 5). Hence, 5 is a minimal element. Similarly, 2 i ...
... Consider the poset (S,≤), where S = {2, 4, 5, 10, 15, 20} and the partial order ≤ is the divisibility relation In this poset, there is no element b ∈ S such that b 5 and b divides 5. (That is, 5 is not divisible by any other element of S except 5). Hence, 5 is a minimal element. Similarly, 2 i ...
Foundations of Mathematics I Set Theory (only a draft)
... part of our book once we know what these objects are). It would be interesting to know what the reader things about the equality 2 = {0, 1}. Does it hold or not? It all depends on the definition of 2. As we will see in the next part, the integer 2 will be defined as the set {0, 1}, so that the equal ...
... part of our book once we know what these objects are). It would be interesting to know what the reader things about the equality 2 = {0, 1}. Does it hold or not? It all depends on the definition of 2. As we will see in the next part, the integer 2 will be defined as the set {0, 1}, so that the equal ...