CMSC 250 FINAL (100% Correct)

2 Implementations of XOR correct answers (x ^ (~y)) ∨ ((~x) ^ y) (x ∨ y) ^ (~(x ^ y)) Carry bit in the half adder correct answers x ^ y How does the full adder add 3 binary digits X, Y, and Z? correct answers 1.) X and Y are added by the 1st HA, producing C1 and St 2.) St and Z are added by the 2nd HA, producing C2 and S 3.) C1 and C2 are added by an OR gate, producing C How does the n bit adder work? correct answers 1.) The first 2 bits are added with a half adder, producing S1 and C1 2.) Each subsequent 2 bits are added with a full adder, with the previous adder providing the carry bit Max value of n bits? correct answers 2^n - 1 B/t S ∋ ∅, S ⊇ ∅, S ∋ {∅}, S ⊇ {∅}, which one is always true? correct answers S ⊇ ∅ |P(A)| correct answers 2^|A| |A U B| correct answers |A| + |B| - |A ⋂ B| Rule for figuring out subsets correct answers For each element in the subset, remove 1 bracket level. If that exact element exists in the supposed superset for all elements, then the subset relation is correct Unique Prime Factorization correct answers (N ∋ x)(P ∋ pk)(N ∋ ak)[x = p1^a1 * p2^a2 * ... * pn^an] General form of Euclidean Proof correct answers 1.) Assume statement is false 2.) Use def of Q to write val in the form a/b 3.) Isolate a^n and clear radicals 4.) Use lemmas to use a fact about a^n to prove something about a 5.) Use 4.) to make a substitution 6.) Isolate b^n 7.) Use lemmas to use a fact about b^n to prove something about b 8.) Use both substitutions to prove the original val is rational, thus making a contradiction Necessity for doing induction problems with sequences correct answers You must first show that from k c = k + 1 c + 1. This allows you to another definition of an that only applies starting at c + 1 Can contrapositives be assumed/supposed? correct answers No