\ (R . (R\A)n (R\B) R\ (AU B). Remark: In this problem, R can be replaced with any set X: your proof need not use anything spe bout R. I write R just to make the problem more concrete. Hint: Draw some pictures. give a short reason why the following statements are true. X 0 is an en ise 3: Either find a counterexample, or (a) If x is real and x2 < 0, then a < 0. Po sitive.ee caus 8 becau any #syvar statenel Potesis Prorosed Joustu te v'a (b) For all integers n, there exists an integer m such that for all integers k, k divides n + m. falr D U (c) For all integers n, there exists an integer m such that for all integers (k) n + m divides k. 10 (d) There is an integer n such that for all integers m there is an integer k such that k divides n +m This C ej The product of a rational number and an irrational number is irrational. FalSe true is an If the product of two irrentienat 's tnis is trve f e he teo o #S t te dete onal numbers is irrational, then so is the sum. becaure Crdless 0 zeros uultiplred, e oilget CO ,an gndlesS peisal, f e acd ise 4: In each of the following, compute f + g, f.g, f/g, and fog, stating the domain of each of the (a) f(x)= 0, g(x) 1/x (b) f(x) r, g(x) 2x +5 (e) f(a)= 1/r, g(a) = (z - 2)/ ) f(z) V1-x, g(a) 2 cos x K 11 ntm eA 3 5: Hard] If a, b, n are integers and a2 +b2 = n, then the remainder of n upon division by 4 is not : [Challenge] Suppose that you have a covering of a sphere by finitely many (closed) hemispher e that there is a subcollection of less than five hemispheres that cover the sphere. hemisphere t of Tn tho o ndo to

Question

Either find a counterexample or give a short reason why: 

b) for all integers n, there exists an integer m suck that for all integers k, k divides n+m. 

c) for all integers n, there exists an integer m suck that for all integers k, n+m divides k. 

d) there is an integer n such that for all integers m there is an integer k suck that k divides n+ m. 

 

 

 

\
(R
.
(R\A)n (R\B) R\ (AU B).
Remark: In this problem, R can be replaced with any set X: your proof need not use anything spe
bout R. I write R just to make the problem more concrete.
Hint: Draw some pictures.
give a short reason why the following statements are true.
X 0 is an en
ise 3: Either find a counterexample, or
(a) If x is real and x2 < 0, then a < 0.
Po sitive.ee caus 8 becau any #syvar
statenel Potesis Prorosed
Joustu te
v'a
(b) For all integers n, there exists an integer m such that for all integers k, k divides n + m. falr
D
U
(c) For all integers n, there exists an integer m such that for all integers (k) n + m divides k.
10
(d) There is an integer n such that for all integers m there is an integer k such that k divides n +m
This
C
ej The product of a rational number and an irrational number is irrational. FalSe
true
is an
If the product of two irrentienat 's tnis is trve f e he teo o #S
t te dete onal numbers is irrational, then so is the sum.
becaure
Crdless
0
zeros uultiplred, e oilget
CO
,an gndlesS peisal, f e
acd
ise 4: In each of the following, compute f + g, f.g, f/g, and fog, stating the domain of each of the
(a) f(x)= 0, g(x) 1/x
(b) f(x) r, g(x) 2x +5
(e) f(a)= 1/r, g(a) = (z - 2)/
) f(z) V1-x, g(a) 2 cos x
K
11
ntm
eA 3
5: Hard] If a, b, n are integers and a2 +b2 = n, then the remainder of n upon division by 4 is not
: [Challenge] Suppose that you have a covering of a sphere by finitely many (closed) hemispher
e that there is a subcollection of less than five hemispheres that cover the sphere.
hemisphere t
of
Tn tho o
ndo
to

Image Transcription

\ (R . (R\A)n (R\B) R\ (AU B). Remark: In this problem, R can be replaced with any set X: your proof need not use anything spe bout R. I write R just to make the problem more concrete. Hint: Draw some pictures. give a short reason why the following statements are true. X 0 is an en ise 3: Either find a counterexample, or (a) If x is real and x2 < 0, then a < 0. Po sitive.ee caus 8 becau any #syvar statenel Potesis Prorosed Joustu te v'a (b) For all integers n, there exists an integer m such that for all integers k, k divides n + m. falr D U (c) For all integers n, there exists an integer m such that for all integers (k) n + m divides k. 10 (d) There is an integer n such that for all integers m there is an integer k such that k divides n +m This C ej The product of a rational number and an irrational number is irrational. FalSe true is an If the product of two irrentienat 's tnis is trve f e he teo o #S t te dete onal numbers is irrational, then so is the sum. becaure Crdless 0 zeros uultiplred, e oilget CO ,an gndlesS peisal, f e acd ise 4: In each of the following, compute f + g, f.g, f/g, and fog, stating the domain of each of the (a) f(x)= 0, g(x) 1/x (b) f(x) r, g(x) 2x +5 (e) f(a)= 1/r, g(a) = (z - 2)/ ) f(z) V1-x, g(a) 2 cos x K 11 ntm eA 3 5: Hard] If a, b, n are integers and a2 +b2 = n, then the remainder of n upon division by 4 is not : [Challenge] Suppose that you have a covering of a sphere by finitely many (closed) hemispher e that there is a subcollection of less than five hemispheres that cover the sphere. hemisphere t of Tn tho o ndo to

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