Let a, b E Z. Prove that a b = gcd(a, b). lcm(a, b). .

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Prove or Disprove: For all real valued functions f, if f(n) is O(2¹), then f(2m) is 0(2m).

Let Express in terms of 1 A = [₂ 2 and 2-² = [2] Α B-¹ 3 and find all its entries. ((BA)T)-¹ A and B-¹

not handwritten   Use K-maps to simplify function f and it’s don’t care condition below. Perform the reduction for (a) the sum-of-products form and (b) the product-of-sums form.f(w, x, y, z) = ∑ m(2, 8, 10, 11) + ∑ d(0, 9)

1. Given the following Minterm Expansion. Plot the terms on a suitable K-Map and then (a) find out the essential prime implicants(if any) and then (b) determine the minimum SOP for the function. You may need additional prime implicants. F(a, b, c) = Im(0, 1, 2, 4, 6)

Use the pumping lemma to prove the following language is nonregular:L = {anb3n} = {abbb, aabbbbbb, aaabbbbbbbbb, …}

Let (a1, b1) and (a2, b2) be open intervals (a1 < b1 and a2 < b2). Show that the Cartesian product (a1,b1) × (a2, b2) is an open region.