Download Discrete Structures Exam 1 for CMSC 203, Fall 2008 and more Exams Discrete Structures and Graph Theory in PDF only on Docsity! Fall 2008 CMSC 203 Discrete Structures Exam 1 1. Circle T for True or F for False as they apply to the following statements: T F The negation of a Conjunctive statement is a Disjunctive statement . T F The negation of an Existential statement is a Universal statement. T F Non-empty sets have at least 2 subsets. T F If F:A → B is function, then F = A × B. T F For any sets A and B, (A ∪ B) ⊆ (A ∩ B). T F The empty set has no subsets. T F N ⊆ Z ⊆ Q ⊆ R ⊆ C. T F Finite sets are Countable. T F That All cars have steering wheels and A motorcycle does not have a steering wheel implies A motorcycle is not a car is an example of Universal Modus Tollens. T F If s is a string over an alphabet Σ, and s is in Σn, then the density of s is n. 2. Find the truth table for the compound statement: [( p ⊕ r) ∨ q] → [p ∧ (¬r → q)] 3. Find the related forms of: For all Integers, x, if x is even, then x is a multiple of 2. CONVERSE: ___________________. NEGATION:_____________________________. 4. Circle the appropriate response when the given argument illustrates Universal Modus Ponens (MP), Universal Modus Tollens (MT), Converse Error (CE), or Inverse Error (IE), : MP MT CE IE All cats eat mice and Tiger is a cat therefore Tiger eats mice. MP MT CE IE All cats eat mice and Tiger is not a cat therefore Tiger does not eat mice. MP MT CE IE All cats eat mice and Tiger does not eat mice therefore Tiger is not a cat. MP MT CE IE All cats eat mice and Tiger eats mice therefore Tiger is a cat. 5. Show that the function f : Z → Z defined as f(x) = (x − 3) is a bijection. 6. Calculate the following (assuming all strings are from the alphabet {0, 1}): (a) |{1, 2, 3} × {4, 5, 6, 7}| (b) d(10010101011001) (c) H(001001100011 , 100101010010) (d) 7. (a) Let f = {(0, 1), (1, 2), (2, 3), (3, 4), (4, 0)}. Find f ° f ° f ° f . f ° f ° f ° f = {(0,___), (1,___), (2,___), (3,___), (4,___)} (b) Find the Inverse of the function of f . f −1 = {(0,___), (1,___), (2,___), (3,___), (4,___)} 8. Use the logic of valid arguments to determine whether or not we can deduce p: p ∨ q ¬t r → s ¬s ∨ t q → r 9. Given a set A and a collection of non-empty subsets of A, {A1, A2, A3, ..., An}, we call this collec- tion a partition of A provided: (i) (A1 ∪ A2 ∪ A3 ∪ ... ∪ An) = A, and (ii) Ai ∩ Aj = ∅ when i ≠ j. (a) What set is partitioned by the collection { {1, 3, 4}, {2, 5}, {7, 8, 9} }? (b) Why is the collection of intervals of the Reals, {... , [−3,−2], [−2,−1], [−1,0], [0,1], [1,2], [2,3], ...} NOT a partition of the Reals? (note: [a,b] = {x ∈ R | a ≤ x ≤ b}. 1.5– 4+( ) 3.7 4+( ) Docsity.com
