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Let $A$ and $B$ be any sets, such that $P(A \cup B) \subseteq P(A) \cup P(B)$,

  1. $A \cup B \in P( A \cup B$)

2.1 Then $A \cup B\in P(A) \cup P( B )$ ( by definition of subset)

2.2 Then $A \cup B\in P(A)$ or $A \cup B\in P( B )$( by definition of or)

2.3 $A\cup B \subseteq A $ or $A\cup B \subseteq B$ ( by definition of power set)

2.4 Suppose $A\cup B \subseteq A $

2.5 $B \subseteq B$

2.6 $B \subseteq A$ $\cup B$ ( by definition of union)

2.7 Let $x\in B$, then we have $x \in A \cup B $ ( by definition of subset)

2.8 Since $x\in A \cup B $ , we have $x\in A$ ( from 2.4 and by definition of subset)

2.9 Therefore we have $B \subseteq A$ ( since $ x\in B$ and $x\in A $)

  1. Similarly, we can prove $A\subseteq B$ using $A\cup B \subseteq B$ .

  2. Therefore, we have prove that if $P(A \cup B ) \subseteq P (A) \cup P(B)$ then either $A\subseteq B$ or $B\subseteq A$

(Part 2.5-2.8 is unecessary, because my professor said that since he did not teach transistivity property of set in his class, we can't directly use it in our proof)

milly
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2 Answers2

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HINT:

Since $A\cup B\in P(A\cup B)\subseteq P(A)\cup P(B)$, we know that either $A\cup B\in P(A)$ or $A\cup B\in P(B)$.

Kenta S
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  • Oh so A U B ⊆ A but i was stuck here: A ⊆ A or B ⊆ A , how do i prove that B ⊆ A in this case? – milly Sep 14 '20 at 12:58
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    If $A\cup B\subseteq A$, since $B\subseteq A\cup B$, we have $B\subseteq A$ by transitivity. – Kenta S Sep 14 '20 at 13:00
  • how do we get B⊆A∪B ? – milly Sep 14 '20 at 13:03
  • Oh from B ⊆ A right? – milly Sep 14 '20 at 13:04
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    @milly The reason why we have $B \subseteq A \cup B$ is because $A \cup B = {x \mid x \in A \lor x \in B}$. If you take arbitrary $x \in B$, by definition of the union, $x \in A \cup B$ – Stokolos Ilya Sep 14 '20 at 13:09
  • @milly $B\subseteq A\cup B$ is automatic, by definition. We cannot use $B\subseteq A$, because this is what we are trying to prove. – Kenta S Sep 14 '20 at 13:10
  • Can you elaborate more on how B⊆A∪B ,becomes B⊆A by transitivity? – milly Sep 14 '20 at 13:12
  • coz we are not supposed to cite transistivity in our exercise(it is not covered in lecture) – milly Sep 14 '20 at 13:13
  • @milly Transitivity is obvious; it is just saying that if $X\subseteq Y$ and $Y\subseteq Z$, then $X\subseteq Z$. In this case, we know that $B\subseteq A\cup B$ and $A\cup B\subseteq A$, so that $B\subseteq A$. – Kenta S Sep 14 '20 at 13:14
  • ok, I will edit my solution and post it afterwards to double confirm again, thanks! – milly Sep 14 '20 at 13:15
  • @KentaS I have edited my solution, could you please spare some time and check again for me?? Appreciated, thank q very much!!! – milly Sep 14 '20 at 13:54
  • you are not "letting" $A\cup B\in P(A\cup B)$. 2.5. why is this necessary? Otherwise, it seems fine.
    • – Kenta S Sep 14 '20 at 13:57
  • @KentaS because I have to use B⊆A ∪B in the following steps but this is not covered in my lecture,so I cannot use it directly, so the only way is to include B⊆B(this one is covered in lecture) and also the union concept to become B⊆A ∪B – milly Sep 14 '20 at 14:04
  • @KentaS i edited the let part(2.), thank q for your help!! – milly Sep 14 '20 at 14:05