set theory

Which of the following sentences are propositions? For those that are, indicate the truth value.

• a. 123 is a prime number.

• b. 0 is an even number.

• c. 푥

• d. Multiply 5푥+2 by 3.

• e. What an impossible question!

State the negation of each of the following statements.

Let 푝: 15 is an odd number. 푞: 21 is a prime number.

Complete the following truth table.

For statements 푝, and 푟, use a truth table to show that each of the following pairs of statements is logically equivalent.

• a. (푝∧푞)⟺푝 and 푝⟹푞.

• b. 푝⟹(푞∨푟) and 푞⟹(푝∨푟).

• c. (푝∨푞)⟹푟 and (푝⟹푞)∧(푞⟹푟).

• d. 푝⟹(푞∨푟) and (푟)⟹(푝⟹푞).

For statements 푝, q, and 푟, show that the following compound statements are tautology.

• a. 푝⟹(푝∨푞).

• b. (푝∧(푝⟹푞))⟹푞.

• c. ((푝⟹푞)∧(푞⟹푟))⟹(푝⟹푟).

Write the contrapositive and the converse of the following conditional statements.

• a. If it is cold, then the lake is frozen.

• b. If Solomon is healthy, then he is happy.

• c. If it rains, Tigist does not take a walk.

Show that (푝∧푞)∧(푝∧푞) is a contradiction.

Find the truth value of each of the following statements: (푝∨푞)∨푟, 푝∨(푞∨푟), 푟⟹(푠∧푝), 푝⟹(푟⟹푠), 푝⟹(푟∨푠), (푝∨푟)⟺(푟∧Ø푠), (푠⟺푝)⟹(Ø푝∨푠), (푞∧Ø푠)⟹(푝⟺푠), (푟∧푠)⟹(푝⟹(Ø푞∨푠)), (푝∨Ø푞)∨푟⟹(푠∧Ø푠).

What can be said about the value of 푝∧푞⟺푝∨푞?