Session 1
1.1 Propositional Logic
Related reading: Stein et al. p. 147–154 or Rosen p. 4–6 and 23–24
Exercise 1.1 Consider (the integers modulo 2). Fill in the addition and multiplication tables below:
Compare these two tables to the truth tables for logic operators (, , , ) in the video above. What is the relationship between the set of truth values and ? Can any of the logic operators be mapped to addition or multiplication?
Exercise 1.2 Let and stand for predicates (constants) that are always false or always true respectively. Simplify the following:
Exercise 1.3 There are two distributive laws. One was proved above. The other is this:
Prove that this law is also true, using truth tables.
1.2 Implication and equivalence
Related reading: Stein et al. p. 155-159 or Rosen p. 6+ and 9+
Exercise 1.4 Show that the expression is equivalent to .
Definition 1 (Equivalence) Given two predicates and . The notation
means the same as
and we say that is equivalent to .
Exercise 1.5 Fill in the truth table for :
Exercise 1.6 Give examples in plain English or Scandinavian where
- «if» appears to mean «if and only if» (or where you think it would for many people).
- where «if» would not mean «if and only if».
- If the sun is shining, then I go swimming.
- If you fail mathematics, then you will not get your degree.
1.3 Direct Proof
Related reading: Stein et al. p. 179–180 or Rosen p. 63–64
Exercise 1.7 Rewrite the following argument in symbolic form, and decide whether or not it is a valid argument.
- If it is Wednesday, then we have fish for dinner.
- It is Wednesday.
- We have fish for dinner.
Exercise 1.8 Rewrite the following argument in symbolic form, and decide whether or not it is a valid argument.
- If it is Wednesday, then we have fish for dinner.
- We have fish for dinner.
- It is Wednesday.
Exercise 1.9 Rewrite the following argument in symbolic form, and decide whether or not it is a valid argument.
- If you fail mathematics, then you will not get your engineering degree.
- You do not get your engineering degree.
- Therefore you did not fail the mathematics module.
Exercise 1.10 Rewrite the following argument in symbolic form, and decide whether or not it is a valid argument.
- If you fail mathematics, then you will not get your engineering degree.
- You fail mathematics
- Therefore you do not get your engineering degree.