Question

ntifiers , 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.

Answer 1

I can only answer 1 question at a time and that too only first four subparts.

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