PDF-Countable and Uncountable Sets In this section we extend the idea of the size of a set

It may come as somewhat of a surprise that there are di64256erent sizes of in64257nite sets At the end of this section we show that there are in64257nitely many di64256erent such sizes For the most part we focus on a classi64257cation of sets into t. J Hildebrand Cardinality Countable and Uncountable Sets Tool Bijections Bijection from of a set Let and be sets A bijection from to is a function that is both injective and surjective Some properties of bijections Inverse functions The inverse func n Otherwise the set is called in64257nite Two sets and are called equinumerous written if there is a bijection A set is called countably in64257nite if We say that is countable if or is 64257nite Example 31 The sets 0 and are equinumerous In Mgr. Lucia . Jureňová. Countable nouns. When the countable noun is Pl. we can use it alone.. I want apples.. We can use some and any.. . . some . - affirmative . and. UNCOUNTABLE NOUNS. and. SOME /ANY / NO / A LOT OF. REVISION ON. REMEMBER. . Countable. . nouns. . are. . nouns. . which. can be . counted. . . and. can be in . the. . singular. . or. Sequences and Summations - vocab. An . arithmetic progression . is a sequence of the form . a, . a+d. , a+2d, … , . a+nd. , …. with fixed a, d in . R . and varying n in . Z. >=0 . A . geometric progression . Form and Usage. countable. / . uncountable. . nouns. countable. . nouns. have. . singular. . and. . plural . forms. we. . can. . use. . numbers. . with. . countable. . nouns. one. . person. www.flickr.com. /photos/. rofi. /2097239111/. Data Structures and . Functional Programming. Computability. Ramin Zabih. Cornell University. Fall 2012. What have we covered?. Tools . for solving difficult computational problems. AND ARTICLES. COUNTABLE NOUNS. Can be singular:. A . job. , a . company. , a . biscuit. Or. plural:. Few. . jobs. , . many. . companies. , . some. . biscuits. .. UNCOUNTABLE NOUNS. Cannot. be plural:. Sostantivi numerabilie non numerabili Uncountable Nouns Uncountable nouns are substances, concepts, etc. that we cannot divide into separate elements . We cannot "count" them. For example, we cannot Cardinality. of Infinite Sets. There be monsters here!. At least serious weirdness!. Cardinality. Recall that the cardinality of a set is merely the number of members in a set. This makes perfect sense for finite sets, but what about infinite sets?. Definition. : The . cardinality. of a set . A. is equal to the cardinality of a set . B. , denoted . . |A| = |. B. |,. if and only if there is a one-to-one correspondence (. Section 2.4. Section Summary. Sequences.. Examples: Geometric Progression, Arithmetic Progression. Recurrence Relations. Example: Fibonacci Sequence. Summations. Special Integer Sequences (. optional. COSC 2001. Lecture. . 5.0. Countable. Countable via a List. Countable via Finite Descriptions. Uncountable. Hierarchy of Infinities. Some . Uncomputable. Problem. 2001: Countable . and Uncountable Infinite. Section. . 2.4. Cardinality. How can we compare the sizes of two sets?. If . S. = {. x.  . .   .  . : . x. 2. = 9}, then . S.  = {–3,.  . 3} and we say that . S. has two elements..