Why does the inequality stand if there are no common elements?

In summary, the conversation discusses the relationship between power sets and unions. The statement $\mathcal{P}A\cup\mathcal{P}B\subset\mathcal{P}(A\cup B)$ is true, but the equality only holds if $A$ and $B$ have no common elements. This means that for the equality to stand, either $A$ must be a subset of $B$ or vice versa. Otherwise, the statement is not true.
  • #1
evinda
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MHB
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Hi! (Smirk)

$$x \in \mathcal{P}A \cup \mathcal{P} B \rightarrow x \in \mathcal{P}A \lor x \in \mathcal{P}B \rightarrow x \subset A \lor x \subset B \rightarrow x \subset A \cup B \rightarrow x \in \mathcal{P} (A \cup B)$$

So, $\mathcal{P}A \cup \mathcal{P}B \subset P(A \cup B) $.

The equality stands, if $A \cap B=\varnothing$.

Could you explain me why the equality stands, if $A,B$ have no common elements? :confused:
 
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  • #2
evinda said:
So, $\mathcal{P}A \cup \mathcal{P}B \subset P(A \cup B) $.

The equality stands, if $A \cap B=\varnothing$.
You can check that this is not so by picking $A=\{1\}$ and $B=\{2\}$. In fact, $\mathcal{P}A\cup\mathcal{P}B=\mathcal{P}(A\cup B)$ iff $A\subseteq B$ or $B\subseteq A$.
 

FAQ: Why does the inequality stand if there are no common elements?

Why does the inequality stand if there are no common elements?

The inequality may still stand because there are other factors at play that contribute to the inequality. For example, there may be historical or systemic barriers that prevent certain groups from having equal access to resources or opportunities.

Can't we just eliminate the inequality by giving everyone the same opportunities?

While providing equal opportunities is important, it may not be enough to eliminate existing inequalities. Other factors such as discrimination, bias, and privilege can still create disparities even when opportunities are made equal.

If there are no common elements, does that mean the inequality is just a result of individual choices?

No, individual choices are often influenced by societal and environmental factors. For example, someone from a marginalized community may have limited options and opportunities compared to someone from a privileged background, leading to different choices and outcomes.

Why is it important to address inequality if there are no common elements?

Addressing inequality is important because it impacts society as a whole. Inequality can lead to social and economic instability, increased crime rates, and hinder overall progress. Additionally, it is a moral imperative to strive for a more equitable society.

Is it possible for everyone to have equal opportunities and outcomes without any common elements?

While it may be difficult to achieve complete equality without any common elements, it is important to continue striving towards it. We can work towards creating a more inclusive and equitable society by addressing systemic issues and promoting diversity and inclusion in all areas of society.

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