Hand-in @ lecture #8

Author

Jonas Smedegaard

Published

14 April 2026

1.5.24

Identify the error or errors in this argument that supposedly shows that if \(\forall x ( P(x) \lor Q(x)\) is true then \(\forall x P(x) \lor \forall x Q(x)\) is true.

The argument

\[\begin{alignat}{2} & \forall x (P(x) \lor Q(x)) \quad && \text{ Premise} \\ & P(c) \lor Q(c) \quad && \text{ Universal instantiation from (1)} \\ & P(c) \quad && \text{ Simplification from (2)} \\ & \forall x P(x) \quad && \text{ Universal generalization from (3)} \\ & Q(c) \quad && \text{ Simplification from (2)} \\ & \forall x Q(x) \quad && \text{ Universal generalization from (5)} \\ & \forall x (P(x) \lor \forall x Q(x)) \quad && \text{ Conjunction from (4) and (6)} \end{alignat}\]

is errornous at several places:

  • Universal instantiation (2) leads to a particular \(c\) but is wrongly assumed arbitrary in universal generalizations (4 and 6) (from a particular \(c\) follows only existential generalization)
  • A conjunction could be simplified to its components but the premise (1) is a disjunction
  • Last transformation (7( is described as conjunction but the actually applied inference rule is disjunction.

1.6.16

Prove that if \(m\) and \(n\) are integers and \(mn\) is even, then \(m\) is even or \(n\) is even.

The theorem states that \(\forall m \forall n P((mn) \to P(m) \lor P(n))\), where \(P(n)\) is n is an even number.

TODO

2.1.4

Suppose that \(A = {2, 4, 6}\), \(B = {2, 6}\), \(C = {4, 6}\), and \(D = {4, 6, 8 }\). Determine which of these sets are subsets of which other of these sets.

B and C are subsets of A, and C is a subset of D.

2.1.34a

Translate this quantifications into English and determine its truth value: \(\exists x \in \mathbb{R} (x^{3} = -1)\)

Some real number exists, that multiplied by itself thrice equals the number \(-1\).

TODO

2.2.16a

Let A and B be sets. Show that \((A \cap B) \subseteq B\)

TODO