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1. ### Set Theory: Prove the set of complex numbers is uncountable

Yeah, my bet! a, b are real numbers, typo... I got this one. Let C be the set of all complex numbers C={a+bi: a, b are real no.} For any real number r can be mapped to a complex no. by r=r+0i, where r=a and is real no., b=0 is also real no. Let R be set of all real numbers R={r+0i: r...
2. ### Cardinalities of Sets: Prove |(0, 1)| = |(0, 2)| and |(0, 1)| = |(a, b)|

Yeah, my bet! a, b are real numbers I've constructed a linear function f: (0,1)->(0,2) defined by f(x)=2x such that f(1/2)=1, when x=1/2 (mid point of domain), y=1 (mid point of range) This linear function is certainly bijection, therefore |(0,1)|=|(0,2)| But how to prove...
3. ### Cardinalities of Sets: Prove |(0, 1)| = |(0, 2)| and |(0, 1)| = |(a, b)|

How to prove the open intervals (0,1) and (0,2) have the same cardinalities? |(0, 1)| = |(0, 2)| Let a, b be real numbers, where a<b. Prove that |(0, 1)| = |(a, b)| ----------------------------------- |(0,1)| = |R| = c by Theorem ----------------------------------- I know that we...
4. ### Set Theory: Prove the set of complex numbers is uncountable

How to prove the set of complex numbers is uncountable? Let C be the set of all complex numbers, So C={a+bi: a,b belongs to N; i=sqrt(-1)} -------------------------------------------------- set of all real numbers is uncountable open intervals are uncountable...

Thanks!
6. ### Prove set S is countable iff there exists a surjective/injective function

(a) A nonempty set S is countable if and only if there exists surjective function f:N->S (b) A nonempty set S is countable if and only if there exists a injective function g:S->N There are two way proves for both (a) and (b) (a-1) prove if a nonempty set S is countable, then there exists...
7. ### Cardinality Problem: Prove |A| < |N|

Okay, here is what I got so far. There should be two steps that I need to prove to show |S|<|N| step 1) to construct a injective function f:S->N step 2) to prove the function f:S->N is NOT bijection (mainly NOT surjective function) Step 1) I started with trying to contrust a injection f:S->N...
8. ### Cardinality Problem: Prove |A| < |N|

Prove cardinality of every finite nonempty set A is less then cardinality of natural number N |A|<|N| set A is nonempty finite set natural number N is denumerable (infinite countable set) |A|<|N| if there exist a injective (one-to-one) function f: A->N, but NO bijective function, which...