Question

Let ℤ be the set of all integers and let, (20) 0 = ∈ ℤ, 1 = ∈ ℤ, 2 = ∈ ℤ, 3 = = 4 + 3, for some integer . Is {0, 1, 2, 3 } a partition of ℤ? Explain your answer.

Answer 1

`\{0, 1, 2, 3\}` is a partition of Z

Explanation:

Given

`$$A _ { 0 } = \{n \in \mathbf { Z } | n = 4 k$$,` for some integer k
`\}`

`$$A _ { 1 } = \{ n \in \mathbf { Z } | n = 4 k + 1$$,` for some integer k},

`$$A _ { 2 } = { n \in \mathbf { Z } | n = 4 k + 2$$,` for some integer k},

and

`$$A _ { 3 } = { n \in \mathbf { Z } | n = 4 k + 3$$,`for some integer k}.

Required

Is
`\{0, 1, 2, 3\}` a partition of Z

Let

`k = 0`

So:

`$$A _ { 0 } = 4 k`

`$$A _ { 0 } = 4 k \to $$A _ { 0 } = 4 * 0 = 0`

`$$A _ { 1 } = 4 k + 1$$,`

`A _ { 1 } = 4 *0 + 1$$ \to A_1 = 1`

`A _ { 2 } = 4 k + 2`

`A _ { 2} = 4 *0 + 2$$ \to A_2 = 2`

`A _ { 3 } = 4 k + 3`

`A _ { 3 } = 4 *0 + 3$$ \to A_3 = 3`

So, we have:

`\{A_0,A_1,A_2,A_3\} = \{0,1,2,3\}`

Hence:

`\{0, 1, 2, 3\}` is a partition of Z