CS 23022 Discrete Structures for Computer Science Homework ...

[Pages:2]CS 23022 Discrete Structures for Computer Science

Homework 1 Due Date: Thursday 6/20/2013

Q1) What's the negation of each of these propositions?

[4 points]

a) Jennifer and Teja are friends.

b) There are 13 items in baker's dozen.

c) Abby sent more than 100 text messages every day.

d) 121 is a perfect square.

Q2) Let and the propositions

: I bought a lottery ticket this week.

: I won the million dollar jackpot.

Express each of these propositions as English sentences

[16 points]

a)

b)

c)

d)

e)

f) )

g)

h)

( )

Q3) For each of these sentences, determine whether an inclusive or, or an exclusive or, is intended.

Explain your answer.

[4 points]

a) Experience with C++ or Java is required. b) Lunch includes soup or salad. c) To enter the country you need a passport or a voter registration card. d) Publish or perish.

Q4) State the converse, contrapositive and inverse of each of these conditional statements.

a) If it snows tonight, then I will stay at home. b) I go to the beach whenever it's a sunny summer day. c) When I stay up late, it is necessary that I sleep until noon.

[12 points]

Q5) Determine whether these biconditionals are True or false.

[8 points]

a) b) c) d)

Q6) Write each of these statements in the form " if p, then q" in English. a) It is necessary to wash the boss's car to get promoted. b) Winds from south implies a spring thaw.

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[14 points]

CS 23022 Discrete Structures for Computer Science

Homework 1 Due Date: Thursday 6/20/2013

c) A sufficient condition for the warranty to be good is that you bought you're the computer less than a year ago.

d) Willy gets caught whenever he cheats. e) You can access the website only if you pay a subscription fee. f) Getting elected follows from knowing the right people. g) Carols get seasick whenever she is on boat

Q7) Verify the equivalence ( p q) ( p r) p (q r)

a) Using truth tables.

[10 points]

b) using the logical equivalences given in class(table of logical equivalences in textbook pages 27 and 28 ) (without using truth table). [10 points]

Q8) Determine the truth value of each of these statements if the universe of discourse of each variable consists of all real numbers.

a) xy(x2 y)

[10 points]

b) xy( y x2 )

c) xy(xy 0)

d) xy(x y y x)

e) x(x 0 y(xy 1))

f) xy( y 0 xy 1)

g) xy(x y 1)

h) xy(x 2y 2 2x 4y 5)

i) xy(x y 2 2x y 1)

j) xyz(z (x y) / 2)

Q9: Rewrite each of these statements so that all negation symbols immediately precede predicates (that is no negations is outside a quantifier or an expression involving logical connectives). [12 points]

a) xyP(x, y)

b) yxP(x, y) c) yx(P(x, y) Q(x, y)

d) (xyP(x, y) xyQ(x, y))

e) x(yzP(x, y, z) zyP(x, y, z))

f) xyzP(x, y, z)

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