Using Pen and paper answer all the following questions.

4 ) show that the two given sets have equal cardinality by describing a bijection from one to the other. Describe your bijection with a formula ( not as a table ). a ) The set of even integers and the set of odd integers. b) z and S = EXER : sinx = 1} c) 80, 13 x N and z " ) a) prove that the set A = {(min ) ENXN : many is countably infinite. D) Prove or disprove : There exists a bijective function : a-R ( ) Prove or disprove : The set 2 xQ is countably infinite. 6) comparing cardinalities a ) prove or disprove: if A CBSC and A and c are countably infinite , then B is countably infinite b ) Prove or disprove: The set f ( a , , a z, a 3 ... ): a; Ez of infinite sequence of integers is countably infinite.1) a ) suppose A = $ 1, 2 , 3, 4 7 , B = 2 0, 1, 2 9 C = [1, 2, 8 3. let $: A-B be f = { ( 1, 0) , ( 2 , 1) , ( 3, 2 ), (4102 3 , and gi Boc be ge 9 = { co, 1) , ( 1 , 1 ), (2, 3 ) 3. find gof b) consider the functions pig: R -7 R defined as f ( 7 )= - and g ( x ) = 32+ 2 find the formulas for gof and fog . ( ) consider the functions fig: 2x2 7 2x2 defined as i ( min) = (3m-un, 2 m+1 ) and 9( min ) = (5mtn , m ) . Find the formulas for gof and fog 2 ) a ) The function f: RT (0,0) defined as flu)= ent is bijective. find its inverse . b) The function : 2x2 + 2X2 defined by the formula f ( m , n ) = ( 5m+ um , um+3n ) is bijective find its inverse ( ) Is the function o: P (2 ) - P(2 ) defined as o ( x ) = x bijective? If so , find o- 3) a ) consider the function p : $ 1 , 2 , 3 , 4 , 5 , 6 , 7 7 - 8 0, 1 , 2 , 3 , 4, 51 6, 71 8, 9 9 given as I = 8 ( 1 , 3) , ( 2 , 8 ) , ( 3 , 3 ), ( 4 1 1 ) , ( 5 , 2 ) , ( 6 / 4 ) , ( 7 , 6 ) 3 find f ( $ 1 , 2, 3 ) , 8 8 4 , 5, 6, 7 3 , 8 (0 ), f-1(8 0, 5, 94 ) and f-"( 80, 3, 5, 93 ). 6) Given a function f; ASB and subsets w, XSA, then i ( wnx ) = & ( w) of ( x ) is false in general: produce a counter example. ( ) Given P: A - B and subsets y, 2 CB, prove $1( 402 ) = $-'(4 ) nf- (2) D) consider fi AB. prove that I is injective if and only if * = f ( f (x)). for all x SA. Prove that & is swijective if and only if P ( P -1 ( 4 ) ) = 4 for all Y EB