Questions: Universal Formulas and Preservation under Substructures

Question 1

Suppose B is a structure satisfying ∀x ∀y (x + y = y + x) (commutativity). If A is a substructure of B, which statement is correct?

• A. A also satisfies commutativity, because the universal formula is preserved going to substructures
• B. A may or may not satisfy commutativity — substructures can violate properties of the parent
• C. A satisfies commutativity only if A has the same cardinality as B
• D. Whether A satisfies commutativity depends on the specific operation, not on the logical form of the formula

Answer: (A)

Commutativity ∀x ∀y (x + y = y + x) is a universal sentence (universal quantifiers over a quantifier-free matrix). The preservation theorem says: if a universal sentence holds in B and A is a substructure of B, then it holds in A. The argument is direct: every element of A is also in B, and for any a₁, a₂ ∈ A, commutativity holds for a₁ and a₂ in B (since they are also elements of B), and since A inherits the same operation, it holds in A. Option B would be correct for existential sentences, not universal ones.

Question 2

A group G satisfies the identity axiom: ∃e ∀x (e·x = x·e = x). A substructure A ⊆ G is closed under the group operation. Which claim about the identity axiom in A is best supported?

• A. A necessarily satisfies the identity axiom, because universal formulas are preserved under substructures
• B. The identity axiom is existential, so it is preserved upward from substructures to superstructures, not downward — A is not guaranteed to satisfy it from logic alone
• C. A satisfies the identity axiom if and only if A has finite cardinality
• D. The identity axiom is universal and therefore automatically satisfied by A

Answer: (B)

The identity axiom begins with ∃e — it is an existential sentence. Existential sentences are preserved going UPWARD (from substructures to superstructures), not downward. Logic alone does not guarantee that A contains the identity element or any identity at all. In practice, subgroups of a group do contain the group identity (by the subgroup definition), but this is an additional structural requirement, not a consequence of the logical form. Option A is wrong because the identity axiom is existential, not universal. Option D misidentifies the quantifier form.

Question 3

If an existential sentence ∃x φ(x) holds in a structure A and B is a superstructure of A (A ⊆ B), then ∃x φ(x) also holds in B.

• T. True
• F. False

Answer: True

Existential sentences are preserved going upward: if some element a ∈ A satisfies φ(a), and A ⊆ B so a ∈ B as well, then B also contains a witness for ∃x φ(x). The quantifier-free φ is satisfied by a in A because A and B agree on all relations and functions restricted to elements of A (this is what 'substructure' means). So the existential witness carries forward. This is the dual of universal preservation: ∀-sentences go down (to substructures); ∃-sentences go up (to superstructures).

Question 4

If a universal sentence holds in a substructure A, it is expected to also hold in any superstructure B containing A.

• T. True
• F. False

Answer: False

This reverses the direction of preservation. Universal sentences are preserved going DOWNWARD — from B to A — not upward. If ∀x φ(x) holds in B, then it holds in A (any element of A is in B and satisfies φ). But the converse fails: B might contain elements outside A that violate φ, even if all elements of A satisfy it. For example, the sentence ∀x (x = 0) holds in the substructure {0} of the integers, but obviously fails in the integers themselves. Universal preservation is strictly one-directional.

Question 5

Explain why universal formulas are preserved under substructures (going from a larger structure to a smaller one) but existential formulas are not. Use the definition of substructure in your answer.

Model answer: A substructure A of B has A's domain ⊆ B's domain, and all relations and functions on A are restrictions of those on B. For a universal sentence ∀x φ(x): if it holds in B, then for every element b ∈ B, φ(b) holds. Since every a ∈ A is also in B, φ(a) holds, and since φ is quantifier-free (checked only using relations/functions that A inherits from B), it holds in A too. For an existential sentence ∃x φ(x): even if it holds in B via some witness b ∈ B, that witness might be in B but not in A. A has fewer elements, so it may simply not contain the witness. There is no guarantee φ holds for any element of A.

The proof direction matters: the key move for universal sentences is 'every element of A is an element of B, so the universal claim still applies.' For existential sentences, the failure direction is 'B's witness might not be in A.' Students who understand this asymmetry — and why it follows from the definition of substructure — have grasped the core of preservation theory.