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AI & Data Mining/Week 20/Chapter 20 Tutorial - Introduction to Propositional Logic.md

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+1. Which of the following English sentences express a proposition?
+(a) I think therefore I am.
+Proposition
+
+(b) Do as I say, not as I do!
+Command
+
+(c) Whenever the assignment x = y is executed, the value of y remains unaltered.
+Proposition
+
+(d) Write clearly and legibly.
+Command
+
+(e) How do you know your answers are correct?
+Question
+
+1. List the atomic propositions and connectives which appear in the following propositions and write down a well-formed formula for each one.
+
+(a) If it rains then I am going to get wet.
+Atomic Propositions:
+	It rains
+	I am going to get wet
+Connectives:
+	If
+	Then
+Formula: p => q
+
+(b) Increased spending overheats the economy.
+Atomic Propositions:
+	Increased spending 
+	Overheats the economy
+Connectives: 
+	None, implied non-linguistic
+Formula: p => q
+
+(c) Increased spending coupled with tax cuts overheats the economy.
+Atomic Propositions:
+	Increased spending
+	There are tax cuts 
+	Overheated economy
+Connectives:
+	Coupled with
+Formula: p ^ q => r
+
+(d) Overheating economy is a synonym for rise in excess demand.
+Atomic Propositions:
+	Overheating economy
+	Rise in excess demand
+Connectives:
+	is a synonym for
+Formula: p <=> q
+
+(e) Inflation either rises or does not.
+Atomic Propositions:
+	Inflation rises
+	Inflation does not rise
+Connectives:
+	either / or
+Formula: p 
+
+1. Remove as many brackets as possible from the following propositions without altering their meaning (i.e. the truth table).
+
+(a) ((q ⇔((¬r) ∨(s ∧p))) ⇔(q ⇒p))
+(b) (((p ∧(¬q)) ∧r) ∨s)
+(c) ((p ⇒(q ∨r)) ∧(¬(r ⇒s)))
+(d) ((¬(¬(¬(q ∨r)))) ⇔(q ⇔r))
+(e) (p ∨(q ∨r))
+
+1. Decide using truth tables whether each of the following is a tautology, contradiction or contingency.
+
+(a) p ⇒¬p
+
+(b) p ∧q ⇒p
+
+(c) (p ⇒¬p) ∧(¬p ⇒p)