# 7, thank you complete · the proof of Proposition 2.8. 2. Let f be a one-to-one

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# 7, thank you

complete · the proof of Proposition 2.8. 2. Let f be a one-to-one function from A into B with B finite. Show that A is finite. 3. If A and B are finite sets, show that AU B is a finite set. Conclude that the finite union of finite sets is finite. 4. If X is an infinite set and x is in X, show that X ~X(x). 5. Define explicitly a bijection from [0, 1] onto (0, 1) 6. Complete the proof of Proposition 2.11. 7. Let f be a one-to-one function from A into B with B countable. Prove . that A is countable. 8. For m in N, show that N~ N 1,2,....m). 9. If A and B are countable sets, show that A x B is countable. 10. Let A be an uncountable set and let B be a countable subset of A. Show 11. Let 2l be a collection of pairwise disjoint open intervals. That is, members nle in R and any two distinct members of 2l are disjoint. that A B is uncountable.

Explanation / Answer

7.

B is countable hence there is a one to one function from B to N,N is set of natural numbers

Let, g be such a function from B to N

Claim: gof is a one to one function from A to N

(gof)(x)=(gof)(y)

g(f(x))=g(f(y))

g is one to one hence, f(x)=f(y)

f is one to one function, (given)

Hence, x=y

So, gof is an one to one function from A to N . Hence, A is countable

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