Logic and Propositional Equivalences

A proposition is a statement which has a truth value, either true or false. For example, "Rob is the big man now." is a proposition because it could be considered that either that he is the big man now or that he isn't. "Who's the big man now?" is not a proposition- it's a question. "Don't tell her I'm leaving!" is not a proposition either- it's a command. "3+x=5" is not a proposition because it depends on the value of 'x' whereas "3+2=5" is a proposition and it's true.

In logic we use special notation to describe propositions because everyday language is not precise enough:
→ Implies
¬ Not
↔ Biconditional (If and only If)
∨ Or
∧ And
⊕ Exclusive-Or

Let p, q and r be the following propositions
p: You're in the kitchen drinking wine
q: You are coming back today
r: You are going to die

Turn "p → ¬q" in to normal english.
Answer: If you're in the kitchen drinking wine then you're not coming back today.

Turn "p → (¬q ∧ r)" in to normal english.
Answer: If you're in the kitchen drinking wine then you're not coming back today and you're going to die. Remember and use brackets to make precedence clear.

Express "You are going to come back today or you are going to die" with logic notation.
Answer: q ⊕ r (It has to be one or the other)

We can use truth tables to represent propositions. The truth table for p→ q is:
p q p → q
F F T
F T T
T F F
T T T
This shows that p implies q unless q can be false when p is true.

Converse, Contrapositive, and Inverse
Statement: "The home team wins (p) whenever it is raining (q)." (q→ p)
Converse: "If the home team wins, then it is raining." (p→ q)
Contrapositive: "If it is not raining, then the home team does not win." (¬q → ¬p)
Inverse: "If the home team does not win, then it is not raining." (¬p → ¬q)