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...
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...
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)|
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|(0,1)| = |R| = c by Theorem
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I know that we...
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)}
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set of all real numbers is uncountable
open intervals are uncountable...
(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...
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...
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...