6. Let p and q be propositions. The proposition that is false when p is true and q is false and is true otherwise is denoted by

Let p and q be propositions. The implication is the proposition that is false when p is true and q is false and true otherwise. In this implication p is called the hypothesis (or antecedent or premise) and q is called the conclusion (or consequence).

Because implications arise in many places in mathematical reasoning, a wide variety of terminology is used to express . Some of the more common ways of expressing this implication are: “if p, then q”, “p implies q”, “if p, q”, “p only if q”, “p is sufficient for q”, “q if p”, “q whenever p”, “q is necessary for p”.

NOTE:

There are some related implications that can be formed from .(see 8 and 9 questions)

7. Let p and q be propositions. The proposition that is true when p and q have the same truth values and is false otherwise is denoted by

Let p and q be propositions. The biconditional is the proposition that is true when p and q have the same truth values and is false otherwise.

Note that the biconditional is true precisely when both the implications and are true. Because of this, the terminology “p if and only if q” is used for this biconditional. Other common ways of expressing the proposition are: “p is necessary and sufficient for q” and “if p then q, and conversely”.

8. Find the converse of . The proposition is called the converse of .

9. Find the contrapositive of . The contrapositive of is the proposition .

Example. Find the converse and the contrapositive of the implication “If today is Thursday, then I have a test today”.

Solution: The converse is “If I have a test today, then today is Thursday”. And the contrapositive of this implication is “If I do not have a test today, then today is not Thursday”.

10. Find the bitwise OR of the bit strings 1011 0110 and 1110 0110.

1111 0110

11. Find the bitwise AND of the bit strings 1010 1010 and 1100 0001.

1000 0000

12. Find the bitwise XOR of the bit strings 0111 1101 and 1111 0111.(должны быть разные)

1000 1010

10-12 Use logical operations on each two corresponding numbers

13. Construct the truth table for the proposition .

p	q	p->q	-p<->q
0 0 1 1	0 1 0 1	1 1 0 1	0 1 1 0	0 1 0 0

14. Construct the truth table for the proposition .

p	q	r	-q->-r
0 0 0 0 1 1 1 1	0 1 1 0 0 1 1 0	0 1 0 1 0 1 0 1	1 1 1 0 1 1 1 0	1 1 1 0 0 0 0 1

15. Let p, q and r be the propositions “You get an A on the final exam”, “You do every exercise in this book” and “You get an A in this class” respectively. Write the proposition “Getting an A on the final and doing every exercise in this book is sufficient for getting an A in this class” using p, q and r and logical connectives.

(p˄q)->r

16. Let p and q be the propositions “You get an A on the final exam” and “You get an A in this class” respectively. Write the proposition “To get an A in this class, it is necessary for you to get an A on the final” using p, q and logical connectives.

p->q

17. Evaluate the expression .

(11 1010) ˄ (00 0110) = 00 0010

18. Find the implication that is false.

19. Let p and q be the propositions “It is below freezing” and “It is snowing” respectively. Express the proposition as an English sentence.

(It is below freezing or It is snowing) AND (if It is below freezing then It is snowing)

20) Let p and q be the propositions “You miss the final examination” and “You pass the course” respectively. Express the proposition as an English sentence.

If and only if you not miss the final examination You will pass the course

21. A compound proposition is a tautology if

A compound proposition that is always true, no matter what the truth values of the propositions that occur in it, is called a tautology.

22. The propositions p and q are logically equivalent if

Compound propositions that have the same truth values in all possible cases are called logically equivalent.

23. Find the proposition that is a tautology.

24. Which of the following logical equivalences is a distributive law?

Show that the propositions and are logically equivalent. This is the distributive law of disjunction over conjunction.

25. Find the proposition that is logically equivalent to .

26. Find the proposition that is logically equivalent to .

,	De Morgan’s laws

27. A proposition is a contingency if

A proposition that is neither a tautology nor a contradiction is called a contingency.

28. Find a compound proposition involving the propositions p, q and r that is true when p and q are false and r is true, but is false otherwise. –pAND-qANDr

29) Find a compound proposition involving the propositions p, q and r that is false when p is false and q and r are true, but is true otherwise. –(-pANDqANDr)=pOR-qOR-r

30) Find a compound proposition involving the propositions p, q and r that is true when p and q are true and r is false, but is false otherwise.

28-30 выбрать правильный среди вариантов ответа прикинув, построив таблицу истинности

31) Let P(x) be the statement “x spends less than three hours every weekday in class”, where the universe of discourse for x is the set of students. Express the proposition “ ” in English.

There is a student who don’t spends less than three hours every weekday in class (more or equal than 3)

32. Let P(x, y) be the statement “x has taken y”, where the universe of discourse for x is the set of all students in your class and for y is the set of all computer courses at your school. Express the proposition in English.

There is exist a computer course which is taken by all students in your class

33) Let P(x) be the statement “x can speak Kazakh” and let Q(x) be the statement “x knows the computer language Delphi”, where the universe of discourse for x is the set of all students at your university. Express the sentence “There is a student at your university who can speak Kazakh but who doesn’t know Delphi” in terms of P(x), Q(x), quantifiers and logical connectives.

Еx(P(x) ˄ -Q(x))

34) Let S(x, y) be the statement “x + y = x  y”, where the universe of discourse for both variables is the set of integers. Which of the following statements is true? ЕхЕуS(x,y)

35) Let S(x, y) be the statement “x + 3y = 3x – y”, where the universe of discourse for both variables is the set of integers. Which of the following statements is true? y=x/2 АуExS(x,y)

36) Rewrite the statement so that negations appear only within predicates (that is, so that no negation is outside a quantifier or an expression involving logical connectives).

АуЕхP(x,y)

37) Rewrite the statement so that negations appear only within predicates (that is, so that no negation is outside a quantifier or an expression involving logical connectives).

Еу(-P(y) ˄ ЕxS(x,y))

38) Rewrite the statement so that negations appear only within predicates (that is, so that no negation is outside a quantifier or an expression involving logical connectives).

Ау(Ех-P(x,y) ˄ AxR(x,y))

39) Rewrite the statement so that negations appear only within predicates (that is, so that no negation is outside a quantifier or an expression involving logical connectives).

Ey(AxEzP(x,y,z) ˄ ExAz-S(x,y,z))

40) Which of the following statements is true if the universe of discourse for all variables is the set of all integers?

41) Which of the following statements is true if the universe of discourse of each variable is the set of real numbers?

42) When the statement is false?

43) When the statement is true?

44) Let W(x, y) mean that x has visited y, where the universe of discourse for x is the set of all students in your school and the universe of discourse for y is the set of all Web sites. Express the statement by a simple English sentence.

There are two students x and y, x and y visited the same websites

45) Let Q(x, y) be the statement “x has been a contestant on y”. Express the sentence “At least two students from your school have been contestants on Wheel of Fortune” in terms of Q(x, y), quantifiers, and logical connectives, where the universe of discourse for x is the set of all students at your school and for y is the set of all quiz shows on television.

ЕхЕу(x!=y AND Q(x, Wheel…) AND Q(y, Wheel…))

46) List the members of the set {x | x is a negative integer greater than (– 5)}. {-4,-3,-2,-1}

47) List the members of the set {x | x is an integer such that x² = 155}. {пустое мн-во}

48) Find the power set of {0, 1, 2}.

Example. What is the power set of the set {0, 1, 2}?

Solution: The power set P({0, 1, 2}) is the set of all subsets of {0, 1, 2}. Hence, P({0, 1, 2}) = {, {0}, {1}, {2}, {0, 1}, {0, 2}, {1, 2}, {0, 1, 2}}. Note that the empty set and the set itself are members of this set of subsets.

49) Let A = {1, 2, 3, 4, 5}, B = {2, 4, 6, 8} and C = {1, 2, 4}. Which of the following statements is true?

50) Let A = {a, b, c, d}, B = {b, d, e, f, g, h, s} and C = {m, n, b, c, r, d, f}. Find .

{b c d} U B ={ b с d e f g h s}

51) Let A = {0, 1, 2, 3}, B = {x, y, z} and C = {a, b, c}. Find (a,x,a) - 27 таких видов, см. Вопрос 56

52) Let U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} be the universal set, and let A = {5, 1, 3, 6, 7, 8}, B = {1, 2, 4, 6, 7, 5, 9}. Find . = {0}

53) Let U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} be the universal set, and let A = {5, 1, 3, 4, 7, 8}, B = {1, 2, 4, 6, 7, 5, 9}. Find . = {2,6,9}

54) A set A is a proper subset of a set B if (если все элементы А находятся в В(A!=B) && (AcB))

When we wish to emphasize that a set A is a subset of the set B but that , we write and say that A is a proper subset of B.

55) Let A be a set. The power set of A is

Given a set S, the power set of S is the set of all subsets of the set S.

56) Let A and B be sets. The Cartesian product of A and B is

The Cartesian product of A and B, denoted by , is the set of all ordered pairs (a, b) where and . Hence, .

Example. What is the Cartesian product of A = {1, 2} and B = {a, b, c}?

Solution: The Cartesian product is

The Cartesian products and are not equal, unless A =  or B =  or unless A = B.

57) Let A and B be sets. The union of A and B is

The union of the sets A and B, denoted by , is the set that contains those elements that are either in A or in B, or in both. Thus, .

Example. Find the union of the sets {1, 3, 5} and {1, 2, 3}. Solution:

58) Let A and B be sets. The intersection of A and B is

The intersection of the sets A and B, denoted by , is the set containing those elements in both A and B. Thus, .

Example. Find the intersection of the sets {1, 3, 5} and {1, 2, 3}. Solution:

59) Let U = {1, 2, 3, 4, 5, 6, 7, 8} be the universal set, and A = {1, 2, 4, 5, 6, 8}, B = {2, 3, 4, 5, 7, 8}. Find the complement of A.

Let U be the universal set. The complement of the set A, denoted by , is the complement of A with respect to U. In other words, the complement of the set A is . Thus, .

Example. Let A = {a, e, i, o, u} (where the universal set is the set of letters of the English alphabet). Then .

Solution 59: notA={3,7}

60) Let for i = 1, 2, 3, … Find

The union of a collection of sets is the set that contains those elements that are members of at least one set in the collection.

We use the notation to denote the union of the sets

60: {3,4,5,6,7}

61) Let for i = 1, 2, 3, … Find

The intersection of a collection of sets is the set that contains those elements that are members of all the sets in the collection.

We use the notation to denote the intersection of the sets

61: {7,8,9…infinity}

62. Let f be a function from A to B. Then the codomain of f is

If f is a function from A to B, we say that A is the domain of f and B is the codomain of f.

63) Let f be the function that assigns the first three bits of a bit string of length 3 or greater to that string. Then the codomain of f is

{000, 001, 010, 011, 100, 101, 110, 111}

64) Let A = {a, b, c, d, e, g, h} and B = {0, 1, 3, 4, 5} with f(a) = 3, f(b) = 2, f(c) = 4, f(d) = 0, f(e) = 5, f(g) = 1 and f(h) = 3. Find the image of S = {c, d, e, g, h}.

f(S)={0, 1, 3, 4, 5}

65) A function f is said to be injective …

A function f is said to be one-to-one, or injective, iff implies that x = y for all x and y in the domain of f. A function f is said to be an injection if it is one-to-one.

Remark. A function f is one-to-one iff whenever .

66) A function f is said to be surjective …

A function f from A to B is called onto, or surjective, iff for every element there is an element with . A function f is called a surjection if it is onto.

67) The function f is a one-to-one correspondence, or a bijection, if

The function f is a one-to-one correspondence, or a bijection, if it is both one-to-one and onto.

68) Let f and g be the functions from the set of integers to the set of integers defined by f(x) = 5x – 6 and g(x) = 2x + 3. Find the composition of g and f.

g(f(x))=g(5x-6)=2(5x-6)+3=10x-9

69) Let f be the function that assigns to each positive integer its first digit. Find the range of f.

{1,2…9}

70) Find 2,97. = 2 71) Find 2,02. =3 72) Find –3,43. =-4 73) Find –3,43. =-3

74) Find 2/30 + 7/30. =1 75. Find 2/30 +7/30. = 1 76) Find 1/4+ 1/4+1/4. = 2

77) Which of the following functions from {a, b, c, d} to itself is one-to-one?

78) Which of the following functions from R to R is a bijection?

79) Let S = {–2, 1, 2, 5}. Find f(S) if f(x) = (x² + 2)/3. = {1,2,9}

80) Find if and are functions from R to R.=

f(g(x))=f(15x-7)=(15x-7)^3+12

81) Find 3/4  4/9. =0

82) There are 25 mathematics majors and 44 computer science majors at a college. How many ways are there to pick one representative who is either a mathematics major or a computer science major?

25+44=69

83) There are 25 mathematics majors and 44 computer science majors at a college. How many ways are there to pick two representatives, so that one is a mathematics major and another is a computer science major?

25*44

84) How many bit strings of length 6 begin and end with a 0?

2^4

85) How many strings of six English letters that are start with B are there if letters can be repeated?

26^5

86) A drawer contains a dozen brown socks and two dozen black socks, all unmatched. A man takes socks out at random in the dark. How many socks must he take out to be sure that he has at least four socks of the same color? 7

87) A drawer contains two dozen brown socks and a dozen black socks, all unmatched. A man takes socks out at random in the dark. How many socks must he take out to be sure that he has at least four brown socks? 16

88) A bowl contains 9 red balls and 9 green balls. A woman selects balls at random without looking at them. How many balls must she select to be sure of having at least five balls of the same color? 9

89) A bowl contains 9 red balls and 9 green balls. A woman selects balls at random without looking at them. How many balls must she select to be sure of having at least five red balls? 14

90) How many bit strings of length 6 begin with a 0? 2^5

91) How many bit strings are there of length five or less? 2^5+2^4+2^3+2^2+2^1+2^0

92) Find a decreasing subsequence of maximal length in the sequence 13, 14, 10, 6, 15, 26, 12, 4, 25, 2.

14,10,6,4,2 (13,10,6,4,2)

93) How many permutations of {1, 2, 3, 4, 5, 6} end with 5? 5!

94) Let S = {1, 2, 3, 4, 5, 6}. How many 4-permutations of S are there? P(6,4)

95) Let A = {a, b, c, d, e}. How many 3-combinations of A are there? C(5,3)

96) Sixty tickets, numbered 1, 2, 3, …, 60, are sold to 60 different people for a drawing. Six different prizes are awarded, including a grand prize (a trip to Moscow). How many ways are there to award the prizes if the person holding ticket 25 wins the grand prize?

P(59,5)

97) Find the coefficient of x⁷y⁶ in (x + y)¹³.

C(13,6)

98) Find the coefficient of x⁴y³ in (3x – 4y)⁷.

C(7,3)*3^4*(-4)^3

99) How many different strings can be made from the letters in MATHEMATICS, using all the letters?

(11!)/(2!*2!*2!)

100) How many solutions are there to the equation where x₁, x₂, x₃ and x₄ are nonnegative integers such that

n=4

r=11-(4+3)=4

C(4+4-1,4)=35

101) A croissant shop has plain croissants, peach croissants, cherry croissants, chocolate croissants, almond croissants, apple croissants and broccoli croissants. How many ways are there to choose a dozen croissants?

C(7+12-1,12)

102) How many ways are there to choose seven coins from a piggy bank containing 150 identical pennies and 90 identical nickels?

C(2+7-1,7)

103) How many different strings can be made from the letters in CORONA, using all the letters?

6!/2!

104) Find P(9, 5). 105) Find C(7, 4).

106) List the ordered pairs in the relation R from A = {0, 1, 2, 3, 4} to B = {0, 1, 2, 3} where (a, b)  R if and only if a + b = 4. {(1,3) (2,2) (3,1) (4,0)}

107) Let R₁ = {(1, 1), (1, 2), (2, 1), (2, 3), (3, 2), (3, 4)} and R₂ = {(1, 2), (1, 3), (2, 4), (3, 1), (3, 2), (3, 3)} be relations from {1, 2, 3} to {1, 2, 3, 4}. Find . {(1,2) (3,2)}

108) Let R = {(a, b), (b, c), (c, a), (d, a)} and S = {(a, b), (b, c), (c, c), (d, c)} be relations on A = {a, b, c, d}. Find .

Example. What is the composite of the relations R and S where R is the relation from {1, 2, 3} to {1, 2, 3, 4} with R = {(1, 1), (1, 4), (2, 3), (3, 1), (3, 4)} and S is the relation from {1, 2, 3, 4} to {0, 1, 2} with S = {(1, 0), (2, 0), (3, 1), (3, 2), (4, 1)}?

Solution: is constructed using all ordered pairs in R and ordered pairs in S, where the second element of the ordered pair in R agrees with the first element of the ordered pair in S. For example, the ordered pair (2, 3) in R and (3, 1) in S produce the ordered pair in S. For example, the ordered pair (2, 3) in R and (3, 1) in S produce the ordered pair (2, 1) in . Computing all the ordered pairs in the composite, we find .

108: {(a,c) (b,c) (c,b) (d,b)}

109) Represent the relation R = {(1, 1), (1, 2), (2, 2), (2, 3), (3, 1)} on {1, 2, 3} with a matrix (with the elements of this set listed in increasing order).

1 1 0

0 1 1

1 0 0