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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 .…
Problem 1. 1. Given two statements S and T : (a) Compare the truth tables of
and of
What can you conclude?
(b) Give the truth table of ¬(S ⇒ T ). Then write ¬(S ⇒ T ) using ¬T .
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,
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.
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
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.
For n ∈ Z, if 6 divides n and 2 divides n, then 12 divides n.