Discrete Mathematics Predicates and Quantified Statements Reading Worksheet-Section 3.1 Answer all questions neatly on another sheet of paper. Please skip lines 1. What is a predicate? Explain the difference between the statement calculus and the predicate calculus. What is the form of a universal statement? 2. 3. 4. What is a counterexample? 5. Find a counterexample to show that this statement is false: Vx EQ+,x2 2x 6. What is the form of an existential statement? 7. Show that the following statement is true: E Z such that x3-x 8. Rewrite the following statements using the symbol V and the variable x a) Every real number is positive, negative, or zero b) All students are eager to learn 9. Rewrite the following statements using the symbol 3 and the variable x a) Some computers are inexpensive b) Some people enjoy solving math puzzles 10. What is the hidden quantifier in each of the following statements? a) If a triangle is equilateral, then it is equiangular c) sinx 1 ll. Let the domain D of be all students in a certain college Let F(x) be "x attends football games" and let B(x) be "x attends basketball games" Rewrite the following statements without using quantifiers, variables, or other symbols. a) 3x E D such that F(x)A B(x) b) Vxe D, F(x) v B(x) c) 3x E D such that ~(F(x) v B(x) 12. Let x be an integer Let P(x) be "x is divisible by 2" and let Q(x) be "x is divisible by 6" Rewrite the following statement in symbolic form using appropriate logical connectives All integers that are divisible by 6 are divisible by 2, but some integers that are divisible by 2 are not divisible by 6