Propositional Logic Questions 1-8 PDF

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Propositional Logic Question 1 

Suppose that the statement p → ¬q is false. Find all combinations of truth values of r and s for which (¬q → r) ∧ (¬p ∨ s) is true.

Question 2 •

If the statement q ∧ r is true, determine all combinations of truth values for p and s such that the statement (q → [¬p ∨ s]) ∧ [¬s → r] is true.

Question 3 

Find all combinations of truth values for p, q and r for which the statement ¬p ↔ (q ∧ ¬(p → r)) is true.

Question 4 •

Is (p → q) → [(p → q) → q] a tautology? Why or why not?

Question 5a •

Show that (p → q) ↔ (q → p) is neither a tautology nor a contradiction. What does that tell you about possible relationships between the truth values of a statement and its converse?

Question 5b •

Suppose ¬[(p → q) ↔ (q → p)] is false. Can p ↔ q have both possible truth values? Explain.

Question 6 •

Show that [(p ∨ q) ∧ (r ∨ ¬q)] → (p ∨ r)] is a tautology by making a truth table, and then again by using an argument that considers the two cases “q is true” and “q is false”.

Question 7 Write each of the following statements, in English, in the form “if p, then q”. • (a) I go swimming on Mondays. Let p = Mondays Let q= I go swimming if it is Monday, then I go swimming • (b) In order to be able to go motorcycling on Sunday, the weather must be good. Let p = Sunday is the day to go motorcycling Let q= the weather must be good if Sunday is the day to go motorcycling, then the weather must be good • (c) Eat your vegetables or you can’t have dessert. Let p = you eat your vegetables Let q= you can have dessert if you eat your vegetables, then you can have dessert If you do not eat your vegetables, then you cannot have dessert • (d) You can ride a bicycle only if you wear a helmet. Let p = wearing a helmet Let q= can ride a bicycle if you are wearing a helmet, then you can ride a bicycle • (e) Polynomials are continuous functions. Let p = it is a continuous fraction Let q= it can be a polynomial if it is a continuous fraction, then it can be a polynomial  (f) A number n that is a multiple of 2 and also a multiple of 3 is a multiple of 6. Let p = a number (n) is a multiple of 2 and also a multiple of 3 Let q= a multiple of 6 if a number (n) is a multiple of 2 and also a multiple of 3, then it is a multiple of 6 • (g) You can’t have any pudding unless you eat your meat. Let p = you eat your meat Let q= you can have pudding if you eat your meat, then you can have pudding • (h) The cardinality of a set is either finite or infinite. Let p = it is a set Let q= the cardinality is either finite or infinite if it is a set, then the cardinality is either finite or infinite

Question 8a Write in English the converse, contrapositive and negation of each statement. (a) If I had $1,000,000, I’d buy you a fur coat. Hypothesis of the implication Let p = If I had $ 1 000 000 Conclusion of the implication Let q= I would buy you a fur coat if p, then q implication p → q is logically the same as (not p) or q The negation of (not p) or q is p and (not q) *** the negation of an implication is not an implication

Converse: conclusion → hypothesis q→p If I buy you a fur coat, then I have $ 1 000 000

Contrapositive: (not conclusion) → (not hypothesis) ¬q → ¬p If I did not buy you a fur coat, then I did not have $ 1 000 000

Negation: hypothesis and (not conclusion) p ∧ (not q) I have $ 1 000 000 and I will not buy you a fur coat

Question 8b Write in English the converse, contrapositive and negation of each statement. (b) If it is not raining and not windy, then I will go running or cycling. Hypothesis of the implication Let p = it is not raining Let q= not windy Conclusion of the implication Let r= I will go running Let s = cycling If p ∧ q , then r ∨ s implication p ∧ q → r ∨ s is logically the same as (not p ∧ q ) or r ∨ s Thus its negation is the negation of (not p ∧ q ) or r ∨ s , which is p ∧ q and (not r ∨ s ). *** the negation of an implication is not an implication

Converse: conclusion → hypothesis r∨s →p∧q If I go running or cycling, then it is not raining and not windy

Contrapositive: (not conclusion) → (not hypothesis) ¬ (r ∨ s) → ¬ (p ∧ q ) If I will not go running or cycling, then it is raining and windy

Negation: hypothesis and (not conclusion) p ∧ q ∧ (not r ∨ s ) it is not raining and not windy and I will not go running or cycling

Question 8c Write in English the converse, contrapositive and negation of each statement. (c) A day that’s sunny and not too windy is a good day for walking on the waterfront. Hypothesis of the implication Let p = the day is sunny Let q= not too windy Conclusion of the implication Let r= it is a good day for walking on the waterfront If p ∧ q , then r implication p ∧ q → r is logically the same as (not p ∧ q ) or r Thus its negation is the negation of (not p ∧ q ) or r , which is p ∧ q and (not r). *** the negation of an implication is not an implication

Converse: conclusion → hypothesis r→p∧q If it is a good day for walking on the waterfront, then the day is sunny and not too windy

Contrapositive: (not conclusion) → (not hypothesis) ¬ r → ¬( p ∧ q) If it is not a good day for walking on the waterfront, then the day is not sunny and too windy

Negation: hypothesis and (not conclusion) p ∧ q and (not r). It is sunny and not too windy today and it is not a good day for walking on the waterfront

Question 8d Write in English the converse, contrapositive and negation of each statement. (d) If 11 pigeons live in 10 birdhouses, then there are two pigeons that live in the same birdhouse. Hypothesis of the implication Let p = 11 pigeons live in 10 birdhouses Conclusion of the implication Let q= 2 pigeons live in the same birdhouse If p , then q implication p → q is logically the same as (not p) or q The negation of (not p) or q is p and (not q) *** the negation of an implication is not an implication

Converse: conclusion → hypothesis q→p If 2 pigeons live in the same birdhouse, then 11 pigeons live in 10 birdhouses

Contrapositive: (not conclusion) → (not hypothesis) ¬q→¬p If 2 pigeons do not live in the same birdhouse , then 11 pigeons do not live in 10 birdhouses

Negation: hypothesis and (not conclusion) p ∧ (not q) 11 pigeons live in 10 birdhouses, and there are not two pigeons that live in the same birdhouse

Question 8e Write in English the converse, contrapositive and negation of each statement. (e)If every domino covers a black square and a white square, then the number of black squares equals the number of white squares.

Hypothesis of the implication Let p = every domino covers a black square Let q= every domino covers a white square Conclusion of the implication Let r= the number of black squares equals the number of white squares If p ∧ q , then r implication p ∧ q → r is logically the same as (not p ∧ q ) or r The negation of (not p ∧ q ) or r is p ∧ q and (not r ) *** the negation of an implication is not an implication

Converse: conclusion → hypothesis r→p∧q If the number of black squares equals the number of white squares, then every domino covers a black square and a white square

Contrapositive: (not conclusion) → (not hypothesis) ¬ r → ¬ (p ∧ q) If the number of black squares does not equal the number of white squares, then every domino does not cover a black square and a white square

Negation: hypothesis and (not conclusion) p ∧ q and (not r ) Every domino covers a black square and a white square and the number of black squares does not equal the number of white squares...