8
1. Numbers and Logic
4. Whic h o f th e followin g statement s ar e tru e fo r al l rea l number s x.
a. I f x (1,2], thenx
2
G (1,4].
b. I f x e (-1,2], thenx
2
G (-1,4].
c. Ifx G (-1,2], thenx
2
G (1,4].
5. a . Wha t i s the lengt h o f a diagona l o f a unit squar e (wit h vertice s (0,0) ,
(0,1), (1,0), (1,1))?
b. Wha t i s the lengt h o f a lon g diagona l o f a uni t cube ?
c. Wha t i s the lengt h o f the longes t diagona l o f a uni t hypercub e i n R
4?
d. Wha t i s the lengt h o f the longes t diagona l o f a uni t hypercub e i n R
n?
6. Prov e tha t \/ 2 i s irrational .
7. Prov e tha t log
1 0
15 is irrational .
8. Fin d infinitel y man y nonempt y set s o f natura l number s
N D Si D S2 D Ss D '
such tha t H^L i S
n
= 0 .
9. Identif y eac h o f th e followin g function s / fro m R t o R a s injectiv e (1-1),
surjective (onto) , neither , o r bot h (bijective) .
a. f(x) = —x.
b. f(x) = x
2.
c. f(x) = sinx .
d. f(x) = e
x.
e. f(x) = x
3
+ x
2.
10. Giv e a counterexampl e t o on e o f the followin g fou r formula s fo r image s
and invers e image s o f set s (th e othe r thre e ar e true) :
f(X, U X2) = f{X x) U f(X2), r\Yi U Y2) = f-\Yi) U r\Y 2),
f(x1 n x2) = f(x1) n f(x2), r\Yi n Y2) = r\Y1) n
/-x(y2).
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