1549901369-Elements_of_Real_Analysis__Denlinger_

704 Appendix C • Answers & Hints for Selected Exercises

Negation: :lx E I 3 Vy E 0 , x + y tJ. E. There is an integer such that no

odd integer added to it yields an even sum.

(b) For every odd integer there is an odd integer that, when added to it,

yields an even sum. True. (Odd +odd =even.)

Negation: :Jx E 0 3 Vy E 0, x + y tJ. E. There is an odd integer such t hat

no odd integer added to it yields an even sum.

(c) There is an integer that, when added to any other integer, yields the
other integer as sum. True: that integer is 0.
Negation: Vy EI, 3x E I 3 x+y =/=-x. There is no integer that, when added
to any other integer, yields the other integer as sum.

(d) For every integer , there is an integer that, when added to it, yields 0

as sum. True: the second integer is t he negative of the first.

Negation: :lx E I 3 Vy E I, x + y =/=-0. There is an integer that has no

"additive inverse."

EXERCISE SET B.l

1. AnB AUB Ac BC
   (a) {4,5} {1, 2,3,4,5,6, 7} {6, 7,8,9,10} {1, 2,3,8,9}

(b) (^0) {1, 2, 3,4,5,6} {4,5,6, 7,8, 9,10} {1,2,3,7, 8,9,10}
(c) [3, 4) (0, 6] (-oo, OJ U [4, + oo) (- oo, 3) U (6, + oo)
(d) [l, 2) (-oo, +oo) [2, +oo) (- oo, 1)
A-B B-A Au (B n C) An (BUG)
(a) {1, 2,3} {6,7} {1,2,3,4,5}
(b) {1, 2,3} {4, 5,6} {1,2,3,4,6}
(c) (0,3) [4,6] (0,5)
(d) (-oo, 1) [2, +oo) (- oo, 2)

3. 
   

   x E (An B)c ¢:? x tJ. (An B) ¢:? '"" (x EA and x EB)
   ¢:? x tJ. A or x tJ. B (by de Morgan's law in logic)
   ¢:? x E Ac or x E Bc ¢:? x E Ac U Bc

   
   


4. 
   

   x E (An B) u (An C) ¢:} x E An B or x E An c
   ¢:? (x EA and x EB) or (x EA and x EC)

   
   

{3,4,5}

{2}

(2,4)

(-1, 2)

¢:? x E A and (x E B or x E C) by the distributive law in logic

¢:? x E A and x E B U C

¢:? x EA n (BU C)