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8. Definition: A set A is finite if there exists a non-negative integer c such that...

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Advance Math

8. Definition: A set A is finite if there exists a non-negativeinteger c such that there exists a bijection from A to {n ? N : n ?c}. (The integer c is called the cardinality of A.)

(a) Let A be a finite set, and let B be a subset of A. Provethat B is finite. (Hint: induction on |A|. Note that our proofcan’t use induction on |B|, or indeed refer to “the number ofelements in B” at all, because we don’t yet know that B isfinite!)

(b) Prove that the union of two disjoint finite sets isfinite.

(c) Prove that the union of any two finite sets is finite.(Hint: A ? B = A ? (B ? A))

Answer & Explanation Solved by verified expert
4.3 Ratings (618 Votes)
Denote c n in N n c 1234c then c c is the cardinality of thefinite set ca let A be a finite set and assume that B is a finite subsetof A If B is empty then B is finite So we assume that B isnonemptySince A is finite there exists a    See Answer
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