Given a function I and a subset A of its domain, let I(A) represent the range of lover the set A; that is, I(A) = {I(x) : x E A}. 12 Chapter 1. The Real Numbers (a) Let f(x) = x2. If A = [0,2] (the closed interval {x E R : ° ~ x ~ 2}) and B = [1,4]' find f(A) and f(B). Does f(A n B) = f(A) n f(B) in this case? Does f(A U B) = f(A) U f(B)? (b) Find two sets A and B for which f(A n B) =1= f(A) n f(B). (c) Show that, for an arbitrary function g : R ---+ R, it is always true that g(A n B) ~ g(A) n g(B) for all sets A, B ~ R. (d) Form and prove a conjecture about the relationship between g(A U B) and g(A) U g(B) for an arbitrary function g.

Step-by-step explanation:

(a)

Using the definition given from the problem

Therefore it is true for intersection. Now for union, we have that

Therefore, for this case, it would be true that .

(b)

1 is not a set.

(c)

To begin with  

Therefore

Now, given an element of it will belong to both sets, therefore it also belongs to , and you would have that

, therefore  .

(d)

To begin with , therefore

3.07*10^-4

move the decimal 4 places to the left

.000307

answers

\3. yes the presentation (i.e. odd numbers being written as differences of two squares) is possible for all odd natural number however the presentation may not be unique. for example,  57=112−82=292−28257=112−82=292−282.

\4. 2 can't be written as a difference of two squares because 4-1=3 and 1-1=0 and the difference of squares grows to integers larger that 3.

can i get some in proving questions 1 and 2?