For each of the following sets A, B prove or disprove whether A ⊆ B and B ⊆ A

Asked Oct 14 '20 at 20:06

Active Oct 14 '20 at 22:32

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Would this proof be sufficient for just proving B ⊆ A alone?

$A = \{x ∈ \Bbb Z : ∃y∈\Bbb Z,x = 5y + 1\},\ $$B = \{x ∈ \Bbb Z : ∃y∈\Bbb Z, x = 10y − 9\}$

Suppose some arbitrary element x is in B. if x∈B, then x∈A by definition of B ⊆ A. Also, by the definition of set B,

x = 10y - 9 for some y = 10y - 9 + 10 -10 = 10y - 10 + 1 = 5(2y-5) + 1

Since, 2y-5 is an integer, x is in A by definition. Therefore B ⊆ A.

edited Oct 14 '20 at 22:22

William Elliot

asked Oct 14 '20 at 20:06

Nicholas

It will be $2y-2$ instead of $2y-5$. Otherwise it is okay. – QED Oct 14 '20 at 20:10
For brevity: Replace your first two lines by: 'Take $x \in B$. Then'. Skip the second line of the 4 equations. Otherwise, fine! – DanLewis3264 Oct 14 '20 at 20:13

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