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Problem 1.      1. Given two statements S and T : (a)  Compare the truth tables of

  • the negation of S ∧ T ,

 

and of

 

  • (¬S) ∨ (¬T ).

 

What  can you conclude?

 

(b)  Give the truth table of ¬(S  ⇒ T ). Then write ¬(S  ⇒ T ) using ¬T .

 

 

  1. Consider the following statement:

 

 

If it is raining then  I will take the bus, and otherwise I will ride my bicycle.

 

(a)  Convert the above statement into propositional calculus, using ∧, ∨, ¬, and  =⇒ .

Be sure to define any statements P, Q, that  you use. (b)  Write  the negation of the above statement,

  1. with symbols
  2. in plain English.

 

Problem 2.  Consider the following two sets of natural numbers.

 

A = {2x − 1 : x ∈ N} = {1, 3, 5, 7, 9, . . .}

 

B = {3x : x ∈ N} = {3, 6, 9, 12, 15, . . .}

 

Give a description  of the following two sets.  A list of the first ten elements followed by . . .

is sufficient.

 

  1. {x ∈ N : (x ∈ A) or (x ∈ B)}

 

  1. {x ∈ N : (x ∈ A) =⇒  (x ∈ B)}

 

  1. {x ∈ N : (x ∈ B) =⇒  (x ∈ A)}

 

  1. {x ∈ N : (x ∈ A =⇒ x ∈ B) and (x ∈ B  =⇒  x ∈ A)}

 

Problem 3.  For x ∈ R, prove the following statement:

 

If |x| > 10 then  x2 + 40 > 14x.

 

 

 

 

Problem 4.  Let a, b, and c be integers.  Consider the statements:

 

P : c divides ab Q: c divides a R: c divides b

  1. Write the statement P  =⇒  Q ∨ R in words.

 

  1. Give an example of integers a, b and c for which the statement in part 1. is false.

 

Problem 5.  Let n ∈ Z. Prove the following claim:

 

 

If 4 divides n − 1, then  n is odd and (−1)

n   1

2       = 1.

 

 

Problem 6.      1. Let n ∈ Z. Prove that  if 5n is even then n is even.

 

  1. Let n ∈ Z. Prove that if 5 divides n and 2 divides n, then 10 divides n.

 

  1. Is the following statement true?

 

 

For n ∈ Z, if 6 divides n and 2 divides n, then  12 divides n.