Question

#7. TRUE/FALSE. Determine the truth value of each sentence  (no explanation required).
________(a)   k in Z  k² + 9 = 0.
________(b)  m, n in N,   5m 2n  is in N.   
________(c)   x in R, if |x − 2| < 3, then |x| < 5.
#8. For each statement,
(i) write the statement in logical form with appropriate variables and quantifiers,
(ii) write the negation in logical form,
and (iii) write the negation in a clearly worded unambiguous English sentence.
(a)  Some athletes earn at least one million dollars per year.
(b)  No drivers are toddlers.
(c)  All distances are positive.

Quantifiers , Counterexamples, Disproof (#9, 15 pts)
#9.  For each statement, state whether the statement is true or false. If false,  explain; provide a counterexample as appropriate or a careful explanation. (If true, no explanation expected)
(a)    n in N,   n+23 ≥n3+8.  
(b)   x in R,  x+23 ≥x3+8.
(c) n in N,   4ⁿ + 1  is prime.
(d)   x, y  in R,  if  |x| < |y|, then x² < xy.  
(e)    m in N such that n in N,  m ≤ n.
(f)     n in N, x  in R  such that  n<x.         
(g)     x  in R such that n in N,  n<x.
 
Applications of Logic (#10, 11, 9 pts)
#10. Write the converse, the inverse, and the contrapositive of the statement
              If Lisa is texting, then Lisa is not driving.
Definitions: A real number r is rational iff integers m and n such that r = m/n and n is nonzero.
The set of all rational numbers is denoted by the symbol Q  (for quotient).
A real number s is irrational iff s is not rational.
#11. Consider the following statement:
         For all real numbers x and y, if the product  xy is irrational, then x is irrational and y is irrational.
(a) Carefully state the contrapositive.
(b) Is the contrapositive true or false?  Explain.
Proofs (#12-15, 33 pts)
Recall the definitions of even,  odd, and multiple of a. (These are used in #13 and #14.)
An integer n is even iff n = 2k for some integer k.
An integer n is odd iff n = 2k + 1 for some integer k.
An integer n is a multiple of a iff n = ak for some integer k. (When a = 2, this is exactly the definition of even.)
#12. Prove carefully, relying on the use of the definitions of even and odd:    
          For any integers p and q, if p is even and q is odd, then  p − 3q + 4 is odd.

Answer 1

(b) Tau, True