Stumped on 2 easy function problems!

  • Thread starter binks01
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  • #1
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Main Question or Discussion Point

The first question I have is simple, but when I attempted it, I got stuck.

I'm trying to prove that if f:X->Y and A & B are subsets of X, that f(A intersect B) is a subset of f(A) intersect f(B).

I started by trying to show set containment, beginning with an arbitrary element in f(A intersect B). However, I cannot figure out how to transition into the right hand side of the problem.

----------------------------

The second question I have is proving that if A and B are finite sets having the same cardinality and f:A->B is one-to-one then f is onto.

I missed class this day and can't figure out what cardinality is by reading the chapter.

Someone please help! =\
 

Answers and Replies

  • #2
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1. If [tex]y \in f(A \cap B)[/tex], then [tex]\exists x \in A \cap B[/tex] such that f(x) = y. Then...

2. The cardinality of a set is its "size". The cardinality of a finite set is the number of elements in it. Since A and B are finite sets having the same cardinality, |A| = |B| = n for some natural number n.
 
  • #3
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How do you use |A| = |B| = n to prove f is onto?
 
  • #4
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How do you use |A| = |B| = n to prove f is onto?
What have you tried so far? What happens if it isn't onto? In other words what happens if there is an element of B that is not mapped to by any element of A?
 
  • #5
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Precise answer depends on used finite set definition.
 

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