2 R is the set of real numbers Let T xR 5 x 11 Show th

2) R is the set of real numbers. Let T = { xR | 5 < x < 11 }.

Show that the set T is not countable.

Hint: Find a function that maps interval (0,1) R into the

interval (5,11 R.

2) R is the set of real numbers. Let T = { xR | 5 < x < 11 }.

Show that the set T is not countable.

Hint: Find a function that maps interval (0,1) R into the

interval (5,11 R.

2) R is the set of real numbers. Let T = { xR | 5 < x < 11 }.

Show that the set T is not countable.

Hint: Find a function that maps interval (0,1) R into the

interval (5,11 R.

f(x)=6x+5 maps (0,1) to (5,11)

f(0)=5, f(1)+11

And f is a linear continuous map so it must take up all values in the interval (5,11) by Intermediate value theorem

Let, f(x)=f(y)

6x+5=6y+5

x=y

So, f is one one and onto

HEnce, f is a bijection from (5,11) to (0,1)

Hence, (5,11) is uncountablebecause (0,1) is