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- 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
- 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
- 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))
- 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)