This document contains the homework assignment for Math 150 in Fall 2020. It includes 4 problems to solve and submit by October 15, 2021 at 11:59pm. Problem 1 involves determining if logic formulas are tautologies using reasoning rather than truth tables. Problem 2 gives definitions and asks to prove statements about formulas using induction on complexity. Problem 3 involves proving statements about logical consequence between sets of formulas. Problem 4 asks to determine the truth of statements involving logical consequence between sets of formulas, providing proofs or counterexamples as needed. Practice problems are recommended to work through before attempting the assigned problems.

1. Math 150 Fall 2020 Homework 1
Due date: Friday, October 15, 2021 by 11:59pm
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Recommendation. Many of the homework problems are close to
problems listed
on the Practice Problems sheet. It is a good idea to try the
problems on Practice
Problem sheet first to become comfortable with the topics.
1. In each of the following cases determine whether a given
formula is a tau-
tology and justify your conclusion. Do not use truth tables; give
an argument
based on reasoning instead. The point here is that it is not
necessary to check all
possible evaluations of sentential letters to decide whether the
formulas below are
tautologies. So try to focus only on those evaluations of
sentential letters which are
crucial. See Practice Problems 1, Problem # 1 for an example of
what is meant by
“argument based reasoning”. Additional examples will be done
in the discussion.
(a) (A → B) → ((B → C) → (A → (B ∧ C))
(b) ((A → B) → (B → C)) → (A → (B ∧ C))
2. Given an L-formula ϕ, denote

2. lp(ϕ) = the number of occurrences of left parentheses in ϕ.
b(ϕ) = the number of occurrences of sentential connectives ∧ ,
∨ , → and ↔ in ϕ.
Give a proof of the following statements. Proceed by induction
on complexity of
formulas.
(a) If ϕ is an L-formula then lp(ϕ) = b(ϕ).
(b) If ϕ is an L-formula built using only sentential connective ∧
and v is an
evaluation of sentential letters then v
̄ (ϕ) = 1 if and only if v(A)
= 1 for all
sentential letter which occur in ϕ.
It is important to take into account parentheses. In the lecture
we said that if ϕ is
an L-formula then we abbreviate ϕ in that we drop writing
certain parentheses, as
they do not have any influence on v
̄ (ϕ). For instance, if ϕ is a
formula as in (b) we
can drop all parentheses in ϕ. However, you cannot make use of
this abbreviations
here, because you work with ϕ from the syntactical point of
view, i.e. you study
them as sequences of symbols. Additionally, in (b) you are
proving that such
abbreviations make sense.
3. Consider sets of L-formulas Σ, Σ′, Γ, Γ′. Prove the following
statements.
(a) If Σ ⊆ Σ′ and Σ |= Γ then Σ′ |= Γ.
(b) Give an example of Σ ⊆ Σ′ and an L-formula σ such that Σ′
|= σ but

3. Σ 6|= σ.
(c) If Γ ⊆ Γ′ and Σ |= Γ′ then Σ |= Γ.
(d) Give an example of Γ ⊆ Γ′ and a set Σ such that Σ |= Γ but Σ
6|= Γ′.
4. Determine whether the following statements are true. If so,
give a proof. If not
give an example of evaluation of sentential letters which
demonstrates it is false.
Here A, B and Ak, k ∈ N are sentential letters.
1
2
(a) ¬A → A |= A
(b) {A0 → B, . . . , An−1 → B} |= (A0 ∨ · · · ∨ An−1) → B
(c) {Ak → B | k ∈ N} |= {(A0 ∨ · · · ∨ An−1) → B | n ∈ N}
(d) {Ak → Ak+1 | k ∈ N} |= Γ where
Γ = {A0 → (A1 ∧ · · · ∧ An) | n ∈ N}
(e) {Ak → Ak+1 | k ∈ N} ∪ {Ak ↔ Bk | k ∈ N} |= Γ where
Γ = {A0 → (B1 ∧ · · · ∧ Bn) | n ∈ N}
As a hint, notice that (c) is an infinite variant of (b), so the
argument for (b) should
also help to approach (c). Also, (e) is just a more complicated
variant of (d), so
understanding (d) should help to approach (e).
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