Question

write the set {-1,1} in builder form

Answer 1

We have to write the set {-1, 1} in set builder form.

Let  A = {-1, 1}.

Case 1:

By comparing with natural numbers, we can see,

-1 = 2 - 3 = 2 × 1 - 3

1 = 4 - 3 = 2 × 2 - 3

Here -1 and 1 are written in the form 2n - 3. So,

A = {x : x = 2n - 3, n ∈ N, 1 ≤ n ≤ 2}

A = {x : x = 2n - 3, n ∈ N, 0 < n ≤ 2}

A = {x : x = 2n - 3, n ∈ N, 1 ≤ n < 3}

A = {x : x = 2n - 3, n ∈ N, 0 < n < 3}

Either these are true, or we can simply write,

A = {2n - 3 : n ∈ N, 1 ≤ n ≤ 2}

A = {2n - 3 : n ∈ N, 0 < n ≤ 2}

A = {2n - 3 : n ∈ N, 1 ≤ n < 3}

A = {2n - 3 : n ∈ N, 0 < n < 3}

Like this we can compare them with other two consecutive natural numbers.

For example, we can see that,

-1 = 4 - 5 = 2 × 2 - 5

1 = 6 - 5 = 2 × 3 - 5

Here -1 and 1 are in the form 2n - 5 where n is either 2 or 3. So,

A = {x : x = 2n - 5, n ∈ N, 2 ≤ n ≤ 3}

A = {x : x = 2n - 5, n ∈ N, 1 < n ≤ 3}

A = {x : x = 2n - 5, n ∈ N, 2 ≤ n < 4}

A = {x : x = 2n - 5, n ∈ N, 1 < n < 4}

Or

A = {2n - 5 : n ∈ N, 2 ≤ n ≤ 3}

A = {2n - 5 : n ∈ N, 1 < n ≤ 3}

A = {2n - 5 : n ∈ N, 2 ≤ n < 4}

A = {2n - 5 : n ∈ N, 1 < n < 4}

Either these are true too. Like this we can make more notations!!!

Case 2:

In case 1 the elements are written in relation with AP. Here in case 2 they're going to be written in relation with GP! This is a simple one!

When we consider -1, 1 as first two terms of a GP, then the GP will continue as,

-1, 1, -1, 1, -1, 1,...

Here the algebraic expression is (-1)ⁿ, and also we get that, if 'n' is odd, then the term will be -1, while if 'n' is even, then the term will be 1.

Anyways, -1 and 1 can be written as (-1)ⁿ, right?!

Hence the set builder form will be,

A = {x : x = (-1)ⁿ, n ∈ Z}

Or

A = {(-1)ⁿ : n ∈ Z}

For any values of 'n', we get only -1 and 1, so there's no problem for n ∈ Z, and we know that the same element can't be repeated in a set, even it is, the element will only be considered as one element.

Case 3:

We can write the elements of the set A as the zeroes of a quadratic equation!

And that equation is  "x² - 1 = 0."

So we can write,

A = {x : x² - 1 = 0}

Case 4:

This is a nice one!!!

The elements of the set are -1 and 1 only, whose "absolute values are equal!"

|1| = 1     ;     |-1| = 1

Hence we can simply write the set A as,

A = {x : |x| = 1}