Mathematics: 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.

i don t see itStep-by-step explanation:...

Let ℤ be the set of all integers and let, (20) 0

IDONTHAVEABRAIN

29.07.2021

$\{0, 1, 2, 3\}$ 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

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