Question

1. Show that if A and B are countable sets, then AUB is countable. 2. Show that if An are finite sets indexed by positive integers, then Un An is countable. 3. Show that if A and B are countable sets, then A x B is countable. 4. Show that any open set in R is a countable union of open intervals. 5. Show that any function on R can have at most countable many local maximals. Us

Answer 1

The notion of equivalence is supposed to lead us to a notion of relative size of sets.

Equivalent sets should have same number of elements (i.e cardinality ). A set A is called finite if A= $\emptyset$ or A is equivalent to the set {1,2,3,....,n} for some n$\in$$\N$$\large \mathbb{N}$; otherwise A is said to be infinite and an infinite set A is said to be countable or countably infinite if A is equivalent to $\large \mathbb{N}$ .

1. show that if A and B AUB is countable nel B are are countable sets, then countable Prof A A and n G: IN B. son! If A and A and B B are are finite finite set set then then AUB is also finite. If A & B be countable & IN A be surjusion. ut h: IN AUB serch that h (m 22 f ((H)/2) if nis odd. – 1 g (1/2) if u is even his susjutine mass. . . AUB is also countable,

- AUB is also countable. B is also contable. 3. If A and B is contou thin AX B is also con Proof w. f: INA I: INB be two bijuline opport ut. uo detine hi INXIN – AXB by h (m,n) (fem), g(n)) Now I have to prove his beigerline. let, K:INX IN - N by kim,n) (2m1) 2 . Kis byartine. thirs it is also bjartin. i AX B is contable. - - - - -- - - - - - - - - - - or for a food a EA ut, Bar{ (a, b) & AXB) BeBS. Sines, Bis contable, each Ba is contable. i Bar Countable union of contable sets. Herer a Bais Contable. A BauBa . AXB is also contable. FA

2. If an are finite sets indexed by will words positive integero, Un An is contable We Prodo siner, each ai ; i. ...D. is constable limite. we can arrange their elements collectively in a malfix. A: a,,, 01, as .... 41, sorom 42: 021 922 23 ..... aan. Az: azi 032 a33 azin. anz ann. An Ani so. Ai is a songe of a map on IN IN . Ai is equivalent to the set IN IN Herer, to a sus set of IN : A; 27 Un An is comfetale. 1S - - - - ---- -

4. Show that any open sit in R is contable union of open intervals. Prof: ut G. Z la ratn} when Id is an open set of IR and is the index set. w €1. Then Ix is an open interval of the collection Go W nt Is Then Jad to such that (x-s, ats) cla There exists a rational no such that - s r cuts TE INI, w us define a funn fin & that assigns CEA) to s (ES) for construction of It's an disjoint, it is ingentine. sines, & is an enumerable set and f is ingeline finn, A is at most enumerable. Herer, G' is a Contable collution. T h e true