0.3 Discrete structures logic  (Page 21/23)

a. “If everyone is happy, then Tom is happy” translates to

∀x [H(x) → H(Tom)].

b. “There are happy people” translates to

∃x H(x)

c. “Not everyone is happy” translates to

∀x ¬[H(x)]

d. ”Some people are happy and some are not happy” translates to

∃x [H(x) ⋀¬H(x)]

17. Which of the following sentences are propositions? What are the truth values of those that are propositions?

a. Richmond is the capital of Virginia.

b. 2 + 3 = 7.

c. Open the door.

d. 5 + 7<10.

e. The moon is a satellite of the earth.

f. x + 5 = 7.

g. x + 5>9 for every real number x .

18. What is the negation of each of the following propositions?

a. Norfolk is the capital of Virginia.

b. Food is not expensive in the United States.

c. 3 + 5 = 7.

d. The summer in Illinois is hot and sunny.

19. Let p and q be the propositions

p : Your car is out of gas.

q : You can't drive your car.

Write the following propositions using p and q and logical connectives.

a. Your car is not out of gas.

b. You can't drive your car if it is out of gas.

c. Your car is not out of gas if you can drive it.

d. If you can't drive your car then it is out of gas.

20. Determine whether each of the following implications is true or false.

a. Your car is not out of gas.

b. If 0.5 is an integer, then 1 + 0.5 = 3.

c. If cars can fly, then 1 + 1 = 3.

d. If 5>2 then pigs can fly.

e. If 3*5 = 15 then 1 + 2 = 3.

21. State the converse and contrapositive of each of the following implications.

a. If it snows today, I will stay home.

b. We play the game if it is sunny.

c. If a positive integer is a prime then it has no divisors other than 1 and itself.

22. Construct a truth table for each of the following compound propositions.

a. p ⋀¬ p

b. ( p ⋁¬ q ) → q

c. ( p → q ) ↔(¬ q →¬ p )

23. Write each of the following statements in the form "if p , then q " in English. (Hint: Refer to the list of common ways to express implications listed in this section.)

a. The newspaper will not come if there is an inch of snow on the street.

b. It snows whenever the wind blows from the northeast.

c. That prices go up implies that supply will be plentiful.

d. It is necessary to read the textbook to understand the materials of this course.

e. For a number to be divisible by 3, it is sufficient that it is the sum of three consecutive integers.

f. Your guarantee is good only if you bought your TV less than 90 days ago.

24. Write each of the following propositions in the form " p if and only if q " in English.

a. If it is hot outside you drink a lot of water, and if you drink a lot of water it is hot outside.

b. For a program to be readable it is necessary and sufficient that it is well structured.

c. I like fruits only if they are fresh, and fruits are fresh only if I like them.

d. If you eat too much sweets your teeth will decay, and conversely.

e. The store is closed on exactly those days when I want to shop there.

25. Use truth table to verify the following equivalences.

a. p ⋀ False ⇔ False

b. p ⋁ True ⇔ True

c. p ⋁ p ⇔ p

26. Use truth tables to verify the distributive law p ⋀ ( q ⋁ r ) ⇔ ( p ⋀ q ) ⋁ ( p ⋀ r ).

27. Show that each of the following implications is a tautology without using truth tables.