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Prim3L0v3r
Prim3L0v3r
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I was having trouble with the following proof: $$ (X ∪ Y ) / (Y ∩ Z)=(X / Y ) ∪ (Y / Z) $$

I managed to prove the L.H.S $ \in $ R.H.S.

Though I didn't succed to do the vice versa. So here's my go at it: $$ (X / Y) \cup (Y / Z) \\ (X \cap Y') \cup (Y \cap Z')$$

At this point I observed there are three possiblities:

  • $ x \in (X \cap Y')$, which implies $(x \in X)\cap(X\notin Y)$. However, if x doesn't belong to Y, it means it doesn't belong to $ Y \cap Z'$ either.
  • $ x \in (Y \cap Z') \to (x \in Y) \cap (x \notin Z)$. Same reasoning here, if it's in $Y$, it cannot be inside $(X\cap Y')$.
  • Though if x belongs to both I didn't know how to appraoch it; obviously an element cannot belong to $A$ and $A'$. Their union is $\emptyset$.

I'd like to know if there's any approach or mind-set with set-theory. I find myself struggling with the reasoning not once in this subject even though it's quite trivial.

EDIT: I would like if someone can post their proof using algebra of sets.

I was having trouble with the following proof: $$ (X ∪ Y ) / (Y ∩ Z)=(X / Y ) ∪ (Y / Z) $$

I managed to prove the L.H.S $ \in $ R.H.S.

Though I didn't succed to do the vice versa. So here's my go at it: $$ (X / Y) \cup (Y / Z) \\ (X \cap Y') \cup (Y \cap Z')$$

At this point I observed there are three possiblities:

  • $ x \in (X \cap Y')$, which implies $(x \in X)\cap(X\notin Y)$. However, if x doesn't belong to Y, it means it doesn't belong to $ Y \cap Z'$ either.
  • $ x \in (Y \cap Z') \to (x \in Y) \cap (x \notin Z)$. Same reasoning here, if it's in $Y$, it cannot be inside $(X\cap Y')$.
  • Though if x belongs to both I didn't know how to appraoch it; obviously an element cannot belong to $A$ and $A'$. Their union is $\emptyset$.

I'd like to know if there's any approach or mind-set with set-theory. I find myself struggling with the reasoning not once in this subject even though it's quite trivial.

I was having trouble with the following proof: $$ (X ∪ Y ) / (Y ∩ Z)=(X / Y ) ∪ (Y / Z) $$

I managed to prove the L.H.S $ \in $ R.H.S.

Though I didn't succed to do the vice versa. So here's my go at it: $$ (X / Y) \cup (Y / Z) \\ (X \cap Y') \cup (Y \cap Z')$$

At this point I observed there are three possiblities:

  • $ x \in (X \cap Y')$, which implies $(x \in X)\cap(X\notin Y)$. However, if x doesn't belong to Y, it means it doesn't belong to $ Y \cap Z'$ either.
  • $ x \in (Y \cap Z') \to (x \in Y) \cap (x \notin Z)$. Same reasoning here, if it's in $Y$, it cannot be inside $(X\cap Y')$.
  • Though if x belongs to both I didn't know how to appraoch it; obviously an element cannot belong to $A$ and $A'$. Their union is $\emptyset$.

I'd like to know if there's any approach or mind-set with set-theory. I find myself struggling with the reasoning not once in this subject even though it's quite trivial.

EDIT: I would like if someone can post their proof using algebra of sets.

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Prim3L0v3r
Prim3L0v3r
  • 81
  • 6

Proofs in set theory.

I was having trouble with the following proof: $$ (X ∪ Y ) / (Y ∩ Z)=(X / Y ) ∪ (Y / Z) $$

I managed to prove the L.H.S $ \in $ R.H.S.

Though I didn't succed to do the vice versa. So here's my go at it: $$ (X / Y) \cup (Y / Z) \\ (X \cap Y') \cup (Y \cap Z')$$

At this point I observed there are three possiblities:

  • $ x \in (X \cap Y')$, which implies $(x \in X)\cap(X\notin Y)$. However, if x doesn't belong to Y, it means it doesn't belong to $ Y \cap Z'$ either.
  • $ x \in (Y \cap Z') \to (x \in Y) \cap (x \notin Z)$. Same reasoning here, if it's in $Y$, it cannot be inside $(X\cap Y')$.
  • Though if x belongs to both I didn't know how to appraoch it; obviously an element cannot belong to $A$ and $A'$. Their union is $\emptyset$.

I'd like to know if there's any approach or mind-set with set-theory. I find myself struggling with the reasoning not once in this subject even though it's quite trivial.