Simple proof

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Homework Statement



Trying to prove some of the basic laws in set theory, and would like any opinions on 1 of my proofs (eg hints on how can I improve it, is it even a valid proof). Thanks in advance.

[tex](A \subseteq B \wedge B \subseteq C) \rightarrow (A \subseteq C)[/tex]


Homework Equations





The Attempt at a Solution



1) [tex]\forall x \in A, x \in B$[/tex] - definition of a subset
2) [tex]\forall x \in B, x \in C[/tex] - definition of a subset
3) [tex]\forall x \in A, x \in C[/tex] - 1, 2
4) [tex]A \subseteq C[/tex] - 3, definition of a subset
 

Answers and Replies

  • #2
EnumaElish
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Your logic is correct.

You could also use proof by contradiction:

Suppose A [itex]\nsubseteq[/itex] C. Then there must be some a in A that is not in C. Since B [itex]\subseteq[/itex] C, a cannot be in B. This contradicts A [itex]\subseteq[/itex] B.
 
Last edited:

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