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 April 27th, 2014, 11:02 AM #1 Newbie   Joined: Apr 2014 From: zagreb Posts: 5 Thanks: 0 Set Theory I'm reading Jech's book. Can someone solve exercises 1.6. 1.7. 2.12 help me to understand lemmae 3.6.-3.10. and 5.2. theorem 3.11. 4.5. 4.8.-Baire category theorem April 28th, 2014, 11:00 AM #2 Newbie   Joined: Apr 2014 From: zagreb Posts: 5 Thanks: 0 ({} є S and (for each x є S) x U {x} є S) We call a set S with the above property inductive. A set T is transitive if x є T implies x subset of T. Exercise 1.6. If X is inductive, then {x є X: x is transitive and every nonempty z subset of x has an є-minimal element} is inductive (t is є-minimal in z if there is no s є z such that s є t) Last edited by raul14; April 28th, 2014 at 11:04 AM. May 22nd, 2014, 12:27 PM #3 Senior Member   Joined: Apr 2014 From: zagreb, croatia Posts: 234 Thanks: 33 Math Focus: philosophy/found of math, metamath, logic, set/category/order/number theory, algebra, topology Theorem 4.5. cardin Every perfect set has cardinality of $\displaystyle R$. Proof. Given a perfect set $\displaystyle P$, we want to find a one-to-one function $\displaystyle F$ from $\displaystyle {0, 1}^\omega$ into $\displaystyle P$. $\displaystyle {0, 1}^\omega$ is equipotent to $\displaystyle {0, 2}^\omega$, given a sequence of 0s and 1s, you just map it to the sequence with 2s instead of 1s. C (the Cantor set) is equipotent to $\displaystyle {0, 2}^\omega$, given a real of the form $\displaystyle \sum_{n=1}^{\infty}\frac{a_n}{3^n}$, where each $\displaystyle a_n$= 0 or 2, map it to the sequence with coresponding 0s and 2s, and so the cardinality of C = $\displaystyle 2^\aleph_0$. Therefore cardinality of $\displaystyle R$>=$\displaystyle 2^\aleph_0$, because C is a subset of reals. As the set $\displaystyle Q$ is dense in reals, every real number r is equal to sup{q element of Q: q

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