Question

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

Answer 1

Answer:

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

Step-by-step 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