Let A and B be a pair of disjoint finite sets. Use induction to prove that if A ≈ m and B ≈ n, then A ∪ B ≈ m + n. Conclude that the union of two finite sets is finite.

discrete math

Find out whether the set of integers Z with respect to addition ± and multiplication O is defined as follows:

10) Identify the mistake in the following “proof" by mathematical induction. (Fake) Theorem: For any positive integer n, all the numbers in a set of n numbers are equal to each other. Proof (by mathematical induction): It is obviously true that all the numbers in a set consisting of just one number are equal to each other, so the basis step is true. For the induction step, let A = {a1, a2, a3, . .., an41} be any set of n +1 numbers. Form two subsets each of size n: B = {a1,a2, A3, . numbers in A except an+1, and C consists of all the numbers in A except a2.) By the induction hypothesis, all the numbers in B equal a1 and all the numbers in C equal a1, since both sets have only n numbers. But every number in A is in B or C, so all the numbers in A equal a1; thus all are equal to each other. .., a,} and C = {a1, az, a4, . .. , an+1}. (Precisely, B consists of all the

Que n > 1, Is). Prove, using mathematical induction that for an integer 1+5+9+... + (4n – 3) = n (2n – 1).