## Transcribed Text

1. Let U be the universe of pets and let F denote furry pets (fur = hair, not feathers or
scales). Let
A = {x|x ∈ F}
B = {Cat, Dog, Goldf ish}
C = {Dog, Bird}
(a) Determine whether each of the following ar true or false. Explain.
(i) Dog ∈ A
(ii) {Dog, Cat} ⊆ B
(iii) Dog ⊆ C
(iv) B ⊆ A
(b) What are B ∪ C and C − A and A¯ ∩ B? (3 answers)
(c) What are all elements of the sets B × C. What is |B × C| ?
2. Let f : B → C and g : C → B be functions with:
f : f(Cat) = Dog, f(Dog) = Dog, f(Goldf ish) = Bird
g : g(Dog) = Cat, g(Bird) = Goldf ish
Answer the following, with explanation.
(a) Is f one-to-one (injective) ? Is g one-to-one?
(b) Is f onto (surjective)? Is g onto?
(c) Is f bijective? Is g bijective?
3. Use Venn Diagrams to determine whether for general sets (A+B)−C = A+ (B −C).
Note: Your solution must include at least 4 diagrams, at least 3 of which must be distinct.
4. Determine the truth value of each of the following. Make sure to explain "It is
true/false because ..."
(a) (∀n)N(n
2 ≥ 5)
(b) (∃n)N(n
2 ≥ 5)
(c) (∀m)N(∃n)N(n
2 ≥ m)
(d) (∃n)N(∀m)N(n
2 ≥ m)
1
5. Consider the following logical sequence.
"I will upgrade if the new phone is much better than mine or if mine breaks."
"My phone didn't break."
"I upgraded, so it follows that the new phone is much better than my previous one." \\
(a) Rewrite this using propositional logic. Dene your variables (there should be 3 of
them)
(b) Is this a valid logical conclusion?
(i) Explain in non-mathematical terms, although your logic should relate to what's stated
above (e.g. "Yes, people only ski in winter. It has to be cold to ski.")
(ii) Then use propositional logic. Your answer should refer to converse or inverse at
some point.
(c) Consider the following statement in propositional logic.
(a → b) ∨ (a ∧ ¬b)
Do not simplify this expression before answering the questions.
(i) Use a truth table to determine its truth value.
(ii) Negate it, distributing the negation so that ¬ does not appear anywhere excet accompanying a variable.
6. Consider the universe of reals R.
(a) What is the truth set of the proposition (x
2 + 4 > 0)?
(b) Use an indirect proof to show that if x
17 − 4x + 4 < 0 then x < 1. Note: You
must give an indirect proof. If you have a direct proof then it can probably be modied
to make an indirect proof.
7. Use induction to show that
∀n ≥ 1 :
1
1 × 2
+
1
2 × 3
+ . . . +
1
n(n + 1) =
n
n + 1

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