#3. Suppose that (a, b) SR is an open interval of real numbers. Give a formula f

Question

#3. Suppose that (a, b) SR is an open interval of real numbers. Give a formula for a bijective function g (a, b) (-3,3), and prove that your function is a bijection. Hint: Try a linear function! Note: The point of this problem is the following. Your function in this problem proves that any open interval (a, b) has the same number of elements as the interval Moreover, your function in the previous problem shows that the interval has the same number of elements as the entire set of real numbers. Taken together, this proves that every open interval contains exactly as many elements as the set R.

Explanation / Answer

FOR A FUNCTION TO BE BIJECTIVE

IT SHOULD BE ONE TO ONE AND ONTO MAPPING

LET US TAKE AN EXAMPLE

f(x)=3x+4

one to one

f(x1)=f(x2)

f(x)=3x+4
f(x1)=f(x2)
3x1+4=3x2+4
3x1=3x2
x1=x2

hence f(x) is one to one function

now coming to onto proof

f(x)=y

then,y=3x+4

3x=y-4

x=(y-4)/3

putting this value of x in f(x) we get

3{(y-4)/3}+4=[3(y-4)+12]/3=[3y-12+12]/3=3y/3=y

now f(x)=y

so it is also onto function

so being one one and onto

it is a bijective function

Related Questions

Navigate

• Browse (All)
• Browse #
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.