# For each of the following sets A,B prove or disprove whether A \subseteq B and B

For each of the following sets A,B prove or disprove whether $A\subseteq B$ and $B\subseteq A$
a) $A=\left\{x\in Z:{\mathrm{\exists }}_{y\in z}x=4y+1\right\}$
$B=\left\{x\in Z:{\mathrm{\exists }}_{y\in z}x=8y-7\right\}$
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Step 1
Given: $A=\left\{x\in Z:{\mathrm{\exists }}_{y\in z}x=4y+1\right\}$
$B=\left\{x\in Z:{\mathrm{\exists }}_{y\in z}x=8y-7\right\}$
To prove or disprove: $A\subseteq B,B\subseteq A$
Step 2
Counter Example:
Reason: Let $5\in A$
$5=4\left(1\right)+1$
But $5=8y-7$
$⇒\frac{3}{2}=y$
$⇒\frac{3}{2}\text{⧸}\in Z$
$⇒5\text{⧸}\in B$
Therefore, $A\text{⧸}\subseteq B$
Step 3
Counter Example:
Reason: Let $40\in B$
$40=8\left(5\right)-7$
But $40=4y+1$
$⇒\frac{39}{4}=y$
$⇒\frac{39}{4}\text{⧸}\in Z$
$⇒\text{⧸}\in A$
Therefore, $B\text{⧸}\subseteq A$