Description
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 .
- 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,
- with symbols
- 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.
- {x ∈ N : (x ∈ A) or (x ∈ B)}
- {x ∈ N : (x ∈ A) =⇒ (x ∈ B)}
- {x ∈ N : (x ∈ B) =⇒ (x ∈ A)}
- {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
- Write the statement P =⇒ Q ∨ R in words.
- 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.
- Let n ∈ Z. Prove that if 5 divides n and 2 divides n, then 10 divides n.
- Is the following statement true?
For n ∈ Z, if 6 divides n and 2 divides n, then 12 divides n.