3.1 Definition of a Group
... especially the quantifiers (“for all”, “for each”, “there exists”), which must be stated in exactly the right order. From one point of view, the axioms for a group give us just what we need to work with equations involving the operation in the group. For example, one of the rules you are used to sta ...
... especially the quantifiers (“for all”, “for each”, “there exists”), which must be stated in exactly the right order. From one point of view, the axioms for a group give us just what we need to work with equations involving the operation in the group. For example, one of the rules you are used to sta ...
http://cc.ee.ntu.edu.tw/~farn/courses/DM/slide/Module-4-countability-gra...
... Cardinality: Formal Definition • For any two (possibly infinite) sets A and B, we say that A and B have the same cardinality (written |A |A|=| |=|B B |) iff there exists a bijection (bijective function) from A to B. • When A and B are finite, it is easy to see that such a function exists iff A and B ...
... Cardinality: Formal Definition • For any two (possibly infinite) sets A and B, we say that A and B have the same cardinality (written |A |A|=| |=|B B |) iff there exists a bijection (bijective function) from A to B. • When A and B are finite, it is easy to see that such a function exists iff A and B ...
equivalence relation notes
... of all triangles in the plane, and the variables x, y and z represent triangles. And in the case of the real numbers example, the set U is R and x, y and z are real numbers. Now, it may seem a little weird using set language like that, and there are uses for working with sets, though mostly for reas ...
... of all triangles in the plane, and the variables x, y and z represent triangles. And in the case of the real numbers example, the set U is R and x, y and z are real numbers. Now, it may seem a little weird using set language like that, and there are uses for working with sets, though mostly for reas ...
yes, x∈L no, x∉L - UC Davis Computer Science
... out its blocks. Define when n | m We only got to here – and then I jumped ahead to defining functions. We’ll take up equivalence classes and quotients next time, as well as properties of functions, like injectivity and surjectivity. ...
... out its blocks. Define when n | m We only got to here – and then I jumped ahead to defining functions. We’ll take up equivalence classes and quotients next time, as well as properties of functions, like injectivity and surjectivity. ...
Chapter 3 Finite and infinite sets
... But the idea is clear without worrying about how the definition goes.) We say that A and B can be matched if there is a matching between them. Other terms used for a matching include bijection and one-to-one correspondence. Now we can say precisely what we mean by the size, or ‘cardinality’ of a set ...
... But the idea is clear without worrying about how the definition goes.) We say that A and B can be matched if there is a matching between them. Other terms used for a matching include bijection and one-to-one correspondence. Now we can say precisely what we mean by the size, or ‘cardinality’ of a set ...
Combinatorics
... We then define subsets {k} that contain only k elements from some given set - for example eight letters from the set {a, b, c, ..., f}, a certain number n of atoms from a set of {N} atoms - and then ask what kind of combinations, variations, or relations are possible between {N} and {k}. Note that w ...
... We then define subsets {k} that contain only k elements from some given set - for example eight letters from the set {a, b, c, ..., f}, a certain number n of atoms from a set of {N} atoms - and then ask what kind of combinations, variations, or relations are possible between {N} and {k}. Note that w ...
Recently Littlewood and Offord1 proved the following lemma Let x1
... sum zL= 1 Ek :Y'k we associate a subset of the integers from 1 to n as follows : k belongs to the subset if and only if ek= +1 . If two sums ~k_,Ekxk and Zx=1E1 . xk are both in I, neither of the corresponding subsets can contain the other, for otherwise their difference would clearly be not less th ...
... sum zL= 1 Ek :Y'k we associate a subset of the integers from 1 to n as follows : k belongs to the subset if and only if ek= +1 . If two sums ~k_,Ekxk and Zx=1E1 . xk are both in I, neither of the corresponding subsets can contain the other, for otherwise their difference would clearly be not less th ...
Section 2.2 Families of Sets
... is called a topology on (or for) the universal set U . So is any collection of subsets of U called a topology? The answer is no. There are three restrictions on a family of subsets in order that they form a topology on U . They are: Definition: A topology J on a set U is any collection of subsets of ...
... is called a topology on (or for) the universal set U . So is any collection of subsets of U called a topology? The answer is no. There are three restrictions on a family of subsets in order that they form a topology on U . They are: Definition: A topology J on a set U is any collection of subsets of ...
Functions -
... This being a course on discrete math, it is often useful to discretize numbers, sets and functions. For this purpose the ceiling and floor functions come in handy. DEF: Given a real number x : The floor of x is the biggest integer which is smaller or equal to x The ceiling of x is the smallest integ ...
... This being a course on discrete math, it is often useful to discretize numbers, sets and functions. For this purpose the ceiling and floor functions come in handy. DEF: Given a real number x : The floor of x is the biggest integer which is smaller or equal to x The ceiling of x is the smallest integ ...
