Abstract Algebra set theory

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  • #1
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Let A, B and C be sets.
Prove that if A[itex]\subseteq[/itex]B[itex]\cup[/itex]C and A[itex]\cap[/itex]B=∅, then A[itex]\subseteq[/itex]C.

My attempted solution:
Assume A[itex]\subseteq[/itex]B[itex]\cup[/itex]C and A[itex]\cap[/itex]B=∅.
Then [itex]\vee[/itex]x (x[itex]\in[/itex]A[itex]\rightarrow[/itex]x[itex]\in[/itex]B[itex]\cup[/itex]x[itex]\in[/itex]c).

I'm not sure where to start and how to prove this. Any help would be greatly appreciated. Thank you.
 

Answers and Replies

  • #3
jbunniii
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Suppose [itex]x \in A[/itex]. The goal is to show that this implies [itex]x \in C[/itex].

Since [itex]x \in A[/itex] and [itex]A \subset B \cup C[/itex], it follows that [itex]x \in B[/itex] or [itex]x \in C[/itex]. Can you exclude one of these possibilities?
 
  • #4
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Since A[itex]\bigcap[/itex]B=∅, x[itex]\in[/itex]A or x[itex]\in[/itex]B.
Thus x[itex]\in[/itex]A, x[itex]\notin[/itex]B and x[itex]\in[/itex]C.
Therefor A[itex]\subseteq[/itex]C.

Is that a good way to show how to exclude the possibility of x[itex]\in[/itex]B?
 
  • #5
jbunniii
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Since A[itex]\bigcap[/itex]B=∅, x[itex]\in[/itex]A or x[itex]\in[/itex]B.
Thus x[itex]\in[/itex]A, x[itex]\notin[/itex]B and x[itex]\in[/itex]C.
Therefor A[itex]\subseteq[/itex]C.

You have the right idea, but the wording is a little unclear. The following is not true: "Since A[itex]\bigcap[/itex]B=∅, x[itex]\in[/itex]A or x[itex]\in[/itex]B."

How about the following:

Suppose [itex]x \in A[/itex]. The goal is to show that this implies [itex]x \in C[/itex].

Since [itex]x \in A[/itex] and [itex]A \subset B \cup C[/itex], it follows that [itex]x \in B[/itex] or [itex]x \in C[/itex]. However, [itex]x[/itex] cannot be in [itex]B[/itex], because if it were, then we would have [itex]x \in A \cap B = \emptyset[/itex], which is impossible. Therefore...
 
  • #6
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Ok I think I get it tell me if I worded this correctly:

Suppose [itex]x \in A[/itex] and [itex]A \cap B = \emptyset[/itex]
Since [itex]x \in A[/itex] and [itex]A \subset B \cup C[/itex], this means that [itex]x \in B[/itex] or [itex]x \in C[/itex]. Consequently, [itex]x[/itex] cannot be in [itex]B[/itex], because if it were, then we would have [itex]x \in A \cap B = \emptyset[/itex], which is impossible. Therefore [itex]x \in C[/itex]. Thus [itex]x \in A[/itex] implies [itex]x \in C[/itex] so it follows that [itex]A \subseteq C[/itex]
 
  • #7
jbunniii
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Ok I think I get it tell me if I worded this correctly:

Suppose [itex]x \in A[/itex] and [itex]A \cap B = \emptyset[/itex]
Since [itex]x \in A[/itex] and [itex]A \subset B \cup C[/itex], this means that [itex]x \in B[/itex] or [itex]x \in C[/itex]. Consequently, [itex]x[/itex] cannot be in [itex]B[/itex], because if it were, then we would have [itex]x \in A \cap B = \emptyset[/itex], which is impossible. Therefore [itex]x \in C[/itex]. Thus [itex]x \in A[/itex] implies [itex]x \in C[/itex] so it follows that [itex]A \subseteq C[/itex]

Looks good to me.
 
  • #8
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ok thank you for your help
 

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