Show that the following four conditions are equivalent : 1. A B 2…

Question

Show that the following four conditions are equivalent :

A $\subset$ B
A – B = $\phi$
A $\cup$B = B
A $\cap$B = A

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Answer

Here, first we will prove (i) $\Leftrightarrow$(ii)
Where, (i) = A $\subset $Band (ii) = A - B $\neq$ $\phi$
Suppose that$\mathrm{A} \subset \mathrm{B}$
Now, we need to prove $A-B \neq \phi$
If possible, let $A-B \neq \phi$
Thus, there exists $X \in A, X \neq B,$but this is impossible as $A \subset B$
$\therefore A-B=\phi$
And $\mathrm{A} \subset \mathrm{B}\Rightarrow\mathrm{A}-\mathrm{B} \neq \phi$
Let suppose that $\mathrm{A}-\mathrm{B} \neq \phi$
Now, we have to prove: $\mathrm{A} \subset \mathrm{B}$
Let $\mathrm{X} \in \mathrm{A}$
It can be concluded that$\begin{equation} X \in B(\text { if } X \notin B, \text { then } A-B \neq \phi) \end{equation}$
Thus, A - B =$\begin{equation} \phi = A \subset B \end{equation}$
$\therefore$(i)$\Leftrightarrow$(ii)
Let us assume that A$\subset$B
To prove: A$\cup$ B = B
$\begin{equation} \Rightarrow B \subset A \cup B \end{equation}$
Let us assume that,$\begin{equation} x \in A \cup B \end{equation}$
$\begin{equation} \Rightarrow x \in A \text { or } x \in B \end{equation}$
Taking case I:$\begin{equation} X \in B \end{equation}$
$\begin{equation} A \cup B=B \end{equation}$
Taking Case II :$\begin{equation} X \in A \end{equation}$
$\begin{equation} \Rightarrow X \in B(A \subset B) \end{equation}$
$\begin{equation} \Rightarrow A \cup B \subset B \end{equation}$
Let A$\cup$ B = B
Let us assume that$\begin{equation} X \in A \end{equation}$
$\begin{equation} \Rightarrow X \in A \cup B(A \subset A \cup B) \end{equation}$
$\begin{equation} \Rightarrow X \in B(A \cup B=B) \end{equation}$
$\begin{equation} \therefore A \subset B \end{equation}$
Thus,(i)$\Leftrightarrow$(iii)
Now, to prove (i) $\Leftrightarrow$(iv)
Suppose that A$\subset$B
It can be observed that$\begin{equation} A \cap B \subset A \end{equation}$
Let$\begin{equation} X \in A \end{equation}$
To show:$\begin{equation} X \in A \cap B \end{equation}$
Since,$\begin{equation} A \subset B \text { and } X \in B \end{equation}$
Therefore,$\begin{equation} \mathrm{X} \in \mathrm{A} \cap \mathrm{B} \end{equation}$
$\begin{equation} \Rightarrow A \subset A \cap B \end{equation}$
$\begin{equation} \Rightarrow A=A \cap B \end{equation}$
Similarly, let us assume that$\begin{equation} \mathrm{A} \cap \mathrm{B}=\mathrm{A} \end{equation}$
Let$\begin{equation} X \in A \end{equation}$
$\begin{equation} \Rightarrow X \in A \cap B \end{equation}$
$\begin{equation} \Rightarrow X \in B \text { and } X \in A \end{equation}$
$\begin{equation} \Rightarrow A \subset B \end{equation}$
$\therefore$(i)$\Leftrightarrow$(ii)
Therefore, proved that (i)$\Leftrightarrow$(ii)$\Leftrightarrow$(iii)$\Leftrightarrow$(iv)

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