Use mathematical induction to prove Theorem 9.3.1.
by mathematical induction.
Suppose a finite set A equals of k distinct mutually disjoint subsets .
PROOF BY INDUCTION:
Let be “If a finite set A is the union of n distinct mutually disjoint subsets then: ”
Basis step: n = 1
Let A be the union of 1 distinct mutually disjoint subset A1
Since A = A1, the two sets need to contain the same number of elements:
Thus P (1) is true.
Let be true, thus if A is the union of k distinct mutually disjoint subsets then:
We need to prove that is true.
Let A be the union of k + 1 distinct mutually disjoint subsets
Let thus B is the union of the first k distinct mutually disjoint sets. Since is true:
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