Question

List all the elements of the following set:
C = {x : x is an integer, $-\dfrac{1}{2}$$<$x$<$$\dfrac{9}{2}$}

Answer 1

Hint: In order to find the correct solution to the question, we need to know that when we have been given any set in set-builder form, like C = {f (x): x \[\in \] something, \[a\le x\le b\]}. Then, f(x) represents the function by using which we find the two elements of the set x \[\in \] something represents what type of values of x are acceptable in f (x) like natural numbers, integers, etc. and \[a\le x\le b\] represents that whatever be the situation, x will only belong between the set [a, b].

Complete step-by-step answer:

In this question, we have been asked to list all the elements of the set
C = {x : x is an integer, $-\dfrac{1}{2}$$<$x$<$$\dfrac{9}{2}$}
Here, we have been given x = f(x) where x is an integer and x lies between \[\dfrac{-1}{2}\] and \[\dfrac{9}{2}\]. Now, we know that integers are 0, \[\pm 1\], \[\pm 2\]…. And we have been given that $-\dfrac{1}{2}$$<$x$<$$\dfrac{9}{2}$. If we convert it into decimal form, we get that – 0.5 < x < 4.5. So, we can say that x can possibly be 0, 1, 2, 3, and 4. Now, we will put the different values of x in f (x), and with that, we will find the elements of set C.
We have been given f(x) = x. So we get,
For x = 0, f (x) = 0
For x = 1, f (x) = 1
For x = 2, f (x) = 2
For x = 3, f (x) = 3
For x = 4, f (x) = 4
Hence, all the elements of set C are {0, 1, 2, 3, 4}.

Note: The possible mistake one can make is by taking the wrong values of x like by not taking x = 0 etc. Also, we could have done this question directly by just writing the values of x as the elements of f (x) as f(x) = x but for better understanding, we did in this way. It is better to convert into decimals and check the possible values of x in that range when compared to fractions.