Lydia buys 5 pounds of apples and 3 pounds of bananas for a total of $8.50. Ari buys 3 pounds of apples and 2 pounds of bananas
2 answers:
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Answer:
a) Countably infinite
b) Countably infinite
c) Finite
d) Uncountable
e) Countably infinite
Step-by-step explanation:
a) Let S the set of integers grater than 10.
Consider the following correspondence:
defined by
for
.
Let's see that the function is one-to-one.
Suppose that f(10+k)=f(10+j) for k≠j. Then k-1=j-1. Thus k-j=1-1=0. Then k=j. This implies that 10+k=10+j. Then the correspondence is injective.
b) Let S the set of odd negative integers
Consider the following correspondence:
defined by
.
Let's see that the function is one-to-one.
Suppose that f(-(2k+1))=f(-(2j+1)) for k≠j. By definition, k=j. This implies that the correspondence is injective.
c) The integers with absolute value less than 1,000,000 are in the intervals A=(-1.000.000, 0) B=[0, 1.000.000). Then there is 998.000 integers in A that satisfies the condition and 999.000 integers in B that satifies the condition.
d) The set of real number between 0 and 2 is the interval (0,2) and you can prove that the interval (0,2) is equipotent to the reals. Then the set is uncountable.
e) Let S the set A×Z+ where A={2,3}
Consider the following correspondence:
defined by
Let's see that the function is one-to-one.
Consider three cases:
1. , then 2k=2j, thus k=j.
2. , then 2k+1=2j+1, then 2k=2j, thus k=j.
3. , then 2k=2j+1. But this is impossible because 2k is an even number and 2j+1 is an odd number.
Then we conclude that the correspondence is one-to-one.
The power series is
g(x) = -2x - (2x)2/2 - (2x)3/3 -(2x)4/4 -.......-(2x)n/n - .....
To deduce the power series of g(x) from the power series for f(x) and identify its radius of convergence
The power series for f(x) is just the geometric series derived from 1/1-y ,setting y=2x.
Its radius of convergence is 0.5
Let,
f(x)= 1/1-2x = 1+ (2x) + (2x)2 + .........+(2x)n......+....
The power series expansion (geometric series),
valid for I2xI < 1 , IxI < 0.5
so, radius of convergence = 0.5
The power series for g(x) is found by integrating term by term the power series of f(x) (upto a constant). The radius of converngence of g(d) is the same as that of f(x) (from general theory) =0.5
Now, g(x) = ln(1-2x)
= -2
=-2
g(x) = -2x - (2x)2/2 - (2x)3/3 -(2x)4/4 -.......-(2x)n/n - .....
is the power series expansion for g(x).
radius of convergence =0.5
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