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Can you verify my proof if it is right?

Let A and B be sets. (a) Show that A is a subset of B if and only if for any set C, one has A U C is a subset of B U C. (b) Show that A is a subset of B if and only if for any set C, one has A intersect C is a subset of B intersect C.

(a)

  • To prove (=>)

  • Given A is a subset of B, this implies A U B = B.

  • Given a set C, A U B U C = B U C.

  • Since A is a subset of B, this implies A U C is a subset of B U C.

  • To prove (<=) Solution 1

  • Given A U C is a subset of B U C , For A U C , x is a member of A U C .

  • For B U C, x is a member of B U C.
  • Suppose x is a member of A but not a member of B, meaning A is not a subset of B.

  • Therefore , A U C is not a subset of B U C. Hence, A must be a subset of B.

  • To prove (<=) Solution 2

  • Given A U C is a subset of B U C, For A U C, x is a member of A U C and for B U C, x is a member of B U C.

  • Suppose C is a disjoint set.
  • Since A U C is a subset of B U C, it follows that A is a subset of B.

(b)

  • To prove (=>)
  • Since A is a subset of B , it follows that A U B = B.
  • A intersect C is a subset of A by definition of intersection. B intersect C is a subset of B by definition of intersection.
  • Let x be member of A and member of C. Since A intersect C is a subset of A, x must be in A.
  • Let x be a member of B and member of C. Since B intersect C is a subset of B, x must be in B.

  • Therefore, A intersect C is a subset of B intersect C.

  • To prove (<=)

  • Since A intersect C is a subset of B intersect C, suppose we let Set C be a subset of B and a subset of A as well.
  • C is a subset of A implies A intersect C is equals to C.
  • C is a subset of B implies B intersect C is equals to C as well.
  • Since A intersect C is equals to B intersect C, A is a subset of B.
Joe
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  • tl/dr applies. Your proof is poorly formatted and thus difficult to read. At least add whitespace to separate individual statements to make it easier to follow, and preferably use MathJax symbols, either in addition to, or instead of, Math/English Pidgeon. – Graham Kemp Sep 09 '14 at 03:46

1 Answers1

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Your proofs are incorrect. You are implicitly assuming what you want to prove.

(a) $(\Rightarrow)$ You want to show that if $x \in A \cup C$, then $x \in B \cup C$ for each set $C$ in some understood universal set $U$.

Since $A \subseteq B$, if $x \in A$, then $x \in B$. Let $x \in A \cup C$. Then $x \in A$ or $x \in C$, so $x \in B$ or $x \in C$. Thus, $x \in B \cup C$. Hence, $A \cup C \subseteq B \cup C$.

$(\Leftarrow)$ You want to show that if for each set $C$ in some understood universal set U, $A \cup C \subseteq B \cup C$, then $A \subseteq B$.

Since $A \cup C \subseteq B \cup C$ for each set $C$, then $A = A \cup \emptyset \subseteq B \cup \emptyset = B$.

(b) $(\Rightarrow)$ You want to show that if $A \subseteq B$, then $A \cap C \subseteq B \cap C$ for each set $C$ in some understood universal set $U$.

Since $A \subseteq B$, if $x \in A$, then $x \in B$. Let $x \in A \cap C$. Then $x \in A$ and $x \in C$, so $x \in B$ and $x \in C$. Thus, $x \in B \cap C$. Hence, $A \cap C \subseteq B \cap C$.

$(\Leftarrow)$ You want to show that if for each set $C$ in some understood universal set U, $A \cap C \subseteq B \cap C$, then $A \subseteq B$.

Since $A \cap C \subseteq B \cap C$ for each set $C$, then $A = A \cap U \subseteq B \cap U = B$.

N. F. Taussig
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