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exer2-11 - MACM 101 Discrete Mathematics I Exercises on...

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MACM 101 — Discrete Mathematics I Exercises on Predicates and Quantifiers. Due: Friday, October 14th (at the beginning of the class) Reminder: the work you submit must be your own. Any collabora- tion and consulting outside resources must be explicitly mentioned on your submission. Please, use a pen. 30 points will be taken off for pencil written work. 1. Determine the truth value of each of these statements if the universe of each variable consists of all real numbers. (a) x y (( x + 2 y = 2) (2 x + 4 y = 5)); (b) x y z ( x 2 + y 2 = z 2 ). 2. Use predicates and quantifiers to express this statement “Every student has chatted with at least one other student in at least one chat group.” 3. Find a counterexample, if possible, to this quantified statement, where the universe for all variables consists of all integers x y ( y 2 = x ) . 4. Rewrite the following statement so that negations appear only within predicates (that is, no negation is outside a quantifier or an expression involving logical connectives)
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