Intro to Discrete maths

Showing that A is a Subset of B To show that A⊆B, show that if x belongs to A then x also belongs to B. Showing that A is

Not a Subset of B To show that A⊈B, find a single x∈A such that x∉B

Among the sets studied in calculus and other subjects are intervals, sets of all the real numbers between two numbers a and b, with or without a and b. If a and b are real numbers with a≤b, we denote these intervals by

[a,b]={x | a≤x≤b} [a,b)={x | a≤x<b} (a,b]={x | a<x≤b} (a,b)={x | a<x<b}

⌈x⌉ is which function?

ceiling function

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

P(S)

Match the sets with a true statement about the Cartesian product of those sets. 1. {1,2} and {3,4} 2. {1,2,3,4} and {3,4,5,6} 3. {4,5,6,7} and {4,5,6,7} 4. {a, e, i, o, u} and {b, g, t, d} 5. {1, 2,3} and {1,2,4}

1. It's cardinality is 4 2. (4,3) is a member 3. (5,5) is a member 4. It's cardinality is 20 5. (2,2) is a member

Match the function from Z to Z with it's description. 1. f(x) =x ^3 2. f(x) = x+1 3. f(x) = ⌊ x/2 ⌋ 4. f(x) = x^2+1

1. One to one, not onto. (Every Integer is mapped to a unique value. The range consists only of perfect cubes. This is smaller than the co domain.) 2. One to one, onto (Every Integer is mapped to a unique value. The range is equal to the co-domain) 3. Not One to one, onto (0 and 1 both map to 0. The range is equal to the co-domain) 4. Not One to one, not onto (-1 and 1 both map to 2. Only positive integers are in the range.)

Match the statement with the task needed to prove it. 1. A ⊆ B 2. A ⊄ B 3. A = B 4. A = Ø 5. A =/ Ø

1. Show that if x ∈ A, then x ∈ B 2. Find a single x ∈ A, such that x ∉ B 3. Show that A ⊆ B and B ⊆ A. 4. Show that A ⊆ Ø 5. Find a single x ∈ A

Match the description to the method. 1. Show that each side of the identity is a subset of the other side. 2. For each possible combination of the atomic sets, show that an element in exactly these atomic sets must either belong to both sides or belong to neither side 3. Start with one side, transform it into the other side using a sequence of steps by applying an established identity

1. Subset method 2. Membership table 3. Apply existing identities

The formulas each define a function with domain and codomain equal to Z. Match to it's range. 1. f(x)=x^3 2. g(x)=|x| 3. h(x)=2x 4. i(x) = x+1

1. The perfect cubes 2. The non-negative integers 3. The evens 4. All integers

Fill in the blanks. Let A and B be nonempty sets. If f is a function from A to B, A is the ____ of the function and B is the _____ of the function. If f(a)=b, b is the image of a and a is the ____ of b. The _____ of f is the set of all images of elements of A.

1. domain 2. co domain 3. pre image 4. range

Which of the following functions from R to R are invertible? 1. f(x) = ^5(square root of) x 2. f(x) = x^2-1 3. f(x) = 2^x 4. f(x) = x^5-1 5. f(x) = 2x+7 6. f(x) = x^4

1. f(x) = ^5(square root of) x 4. f(x) = x^5-1 5. f(x) = 2x+7 They must be one-to-one and onto to be invertible.

Rank the two sets by the size of their intersection, from smallest at the top to largest at the bottom. 1. {1,2,3,4,5},{5,6,7,8,9,10} 2. {1,2,3,4,5},{6,7,8,9,10} 3. {2,3,4,5,6},{5,6,7,8,9,10} 4. {2,3,4,5,6},{2,3,4,5,6} 5. {1,2,3,4,5,6,7,8}, {3,4,5,6,7,8,9,10}

1. {1,2,3,4,5},{5,6,7,8,9,10} 2. {1,2,3,4,5},{6,7,8,9,10} 3. {2,3,4,5,6},{5,6,7,8,9,10} 4. {2,3,4,5,6},{2,3,4,5,6} 5. {1,2,3,4,5,6,7,8}, {3,4,5,6,7,8,9,10}

Assume that ∪ = {1,2,3,4,5,6,7,8,9,10}. Let A={2,4,6,8,10} and B= {7,8,9,10}. Match the terms with the set that results from taking the complement of the set. 1. A ∪ B 2. A ⋂ B 3. A - B 4. B - A

1. {1,3,5} A ∪ B = {2,4,6,7,8,9,10}. Thus, ∪ -(A ∪ B) is {1,3,5} 2. {1,2,3,4,5,6,7,9} A ⋂ B = {8,10}. Thus, ∪ - (A ⋂ B) is {1,2,3,4,5,6,7,9} 3. {1,3,5,7,8,9,10} A-B = {2,4,6}. Thus ∪-(A-B) is {1,3,5,7,8,9,10} 4. {1,2,3,4,5,6,8,10} B - A = {7,9}. Thus, ∪-(B-A) is {1,2,3,4,5,6,8,10}

Match each interval with its set builder description. 1. (0,1) 2. [0,1) 3. (0,1] 4. [0,1]

1. {x | 0 < x <1} 2. {x | 0 =< x <1} 3. {x | 0 < x =<1} 4. {x | 0 =< x =<1}

Match the set to it's power set. 1. Ø 2. {Ø} 3. {{Ø,{Ø}}} 4. {Ø,{Ø}}

1. {Ø} The empty set has one subset, itself. 2. {{Ø},Ø} {Ø} has two subsets, itself and the empty set. 3. {Ø,{Ø,{Ø}}} Note this set has one element, namely {Ø,{Ø}}. Hence, it's power set has two elements, the empty set and the set itself. 4. {Ø,{Ø},{{Ø}},{Ø,{Ø}}} {Ø,{Ø}} has for subsets, itself, the empty set, {Ø}, and {{Ø}}

Let U = {1,2,3,4,5,6,7,8} be a universal set, and the ordering of elements of U has the elements in increasing order. A set A has bit string representation 10110101. Select all that are members of A. 1. 7. 2. 4 3. 2 4. 8 5. 6 6. 3

2. 4 The bit in the fourth position is 1, so 4 ∈ A 4. 8 The bit in the final position is 1, so 8 ∈ A 5. 6 The bit in the sixth position is 1, so 6 ∈ A 6. 3 The bit in the third position is 1, so 3 ∈ A

Which of these contains both the integers -1 and 3? 1. (0,6] 2. [-1, 3] 3. (-1, 3) 4. [-1, 5] 5. (-3, 3] 6. (-1, 3]

2. [-1, 3] 4. [-1, 5] 5. (-3, 3]

Suppose a person deposits $100,000 in a savings account yielding 2% a year with interest compounded annually. Determine how much will be in the account after 10 years by calculating P^10, where P^n is the amount in the account after n years. 1. 119,509.26 2. 283,942.10 3. 124,337.43 4. 121,899.44 5. 148,594.74

4. 121,899.44 After 10 years, the amount is (1.02)^10 * 100,000 = 121,899.44

Which of these is the correct first four terms of the geometic progression with initial term 3 and common ratio 1/2? 1. 2/3, 4/3, 8/3, 16/3 2. 3, 6, 12, 24 3. 1/2, 3/2, 9/2, 27/2 4. 3, 3/2, 3/4, 3/8 5. 3/2, 3/4, 3/8, 3/16

4. 3, 3/2, 3/4, 3/8 According to the definition of a geometic progression, the first four terms are 3, 3*1/2, 3*(1/2)^2, 3*(1/2)^3

Let U = {1,2,3,4,5,6,7,8,9} be the universal set, and let A= {2,3,4,7,8}. Select all that are members of ā. 1. 5 2. 1 3. 8 4. 10 5. 4

1. 5 2. 1 These numbers are not in A, so they are in the complement of A.

Match the logic set builder notation to the set notation. 1. {x | x ∈ A v x ∈ B} 2. {x | x ∈ A ^ x ∈ B} 3. {x | x ∈ A ^ x ∉ B} 4. {x | x ∈ B ^ x ∉ A} 5. {x | x ∈ U ^ x ∉ A}

1. A U B 2. A ⋂ B 3. A - B 4. B - A 5. ā

Match the set identity to it's name. 1. A ⋂ U = A 2. ≡A = A 3. A ⋂ (B ⋂ C) = (A ⋂ B) ⋂ C 4. A ∪ B = B ∪ A 5. A ⋂ (A ∪ B) = A 6. A ⋂ (- over) A = ∅ 7. A ∪ U = U 8. (Line over: A ∪ B) = (-over) A ⋂ (-over) B 9. A ∪ (-over) A = A

1. Identity Law 2. Complementation law 3. Associative law 4. Commutative law 5. Absorption law 6. Complement law 7. Domination Law 8. De Morgan's Law 9. Idempotent Law

Match the set to it's size. 1. Ø 2. {Ø,{Ø}} 3. {Ø,Ø} 4. The set of letters in the english alphabet 5. {a, b, S} where S is the set of letters in the English Alphabet 6. Z

1. 0 The size of the empty set is 0 2. 2 This set contains Ø and {Ø} 3. 1 This set contains one element. Repetition doesn't matter. 4. 26 There are 26 letters. 5. 3 This set contains two letters and one set 6. infinite There are an infinite number of integers

Compute the floor or ceiling functions, as indicated by the notion. 1. ⌊ 3/2 ⌋ 2. ⌈-1/2⌉ 3. ⌈-3/2⌉ 4. ⌊ -3/2 ⌋

1. 1 2. 0 3. -1 4.-2

Let f, g, and h be function from the set of integers to the set of integers defined by f(x)=2x, g(x)=x-3, and h(x)=4x+5. Match them to the equivalents. 1. (f o g)(x) 2. (g o h)(x) 3. (h o f)(x)

1. 2x-6 (fog)(x) = f(g(x)) = f(x-3)=(2(x-3)=2x-6 2. 4x+2 (goh)(x) = g(h(x)) = g(4x+5)=(4x+5)-3 = 4x+2 3. 8x+5 (hof)(x) = h(f(x)) = h(2x) = 4(2x) +5 = 8x +5

Which of these begins the two main parts of a direct proof of A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) ? 1. Assume A ∪ (B ∩ C). We must show (A ∪ B) ∩ (A ∪ C) 2. Assume that A ∪ (B ∩ C) =/ (A ∪ B) ∩ (A ∪ C) 3. We will show that (A ∪ B) ∩ ( A ∪ C) ⊆ A ∪ (B ∩ C) 4. We will show that A ∪ (B ∩ C) ⊆ (A ∪ B) ∩ (A ∪ C)

3. We will show that (A ∪ B) ∩ ( A ∪ C) ⊆ A ∪ (B ∩ C) 4. We will show that A ∪ (B ∩ C) ⊆ (A ∪ B) ∩ (A ∪ C)

Which are correct for all integers n and real numbers x? 1. ⌈x⌉ = n if and only if x < n =< x+1 2. ⌈-x⌉ = -⌈x⌉ 3. ⌈x⌉ = n if and only if n-1<x=<n 4. ⌊- x ⌋ = -⌊ x ⌋ 5. ⌊ x + n⌋ = ⌊ x ⌋ + n

3. ⌈x⌉ = n if and only if n-1<x=<n 4. ⌊- x ⌋ = -⌊ x ⌋ 5. ⌊ x + n⌋ = ⌊ x ⌋ + n

What is a Bijective?

A function f (from set A to B) is bijective if, for every y in B, there is exactly one x in A such that f(x) = y. Example : G maps the students in class to grades. The domain is the set of students, the co-domain is the set of grades. G(Yu)= F, G(Kay)=E, G(Aly)=D, G(Yarra)=C, G(Mitch)=B, G(Feldman)=A No two students share a grade.

A function f from A to B is called onto, or a surjection, if and only if for every element b∈B there is an element a∈A with f(a)=b.

A function f is called surjective if it is onto

A function f is said to be one-to-one, or an injection, if and only if f(a)=f(b) implies that a=b for all a and b in the domain of f.

A function is said to be injective if it is one-to-one.

The floor function is

Assigns to the real number x the largest integer that is less than or equal to x.

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

A×B={(a,b) | a∈A∧b∈B}

Two sets are equal if and only if they have the same elements.

Therefore, if A and B are sets, then A and B are equal if and only if ∀x(x∈A↔x∈B). We write A=B if A and B are equal sets.

Note that [a,b] is called the closed interval from a to b and (a,b) is called the open interval from a to b. Each of the intervals [a,b], [a,b), (a,b], and (a,b) contains

all the real numbers strictly between a and b. The first two of these contain a and the first and third contain b

The set A is a subset of B, and B is a superset of A, if and only if every element of A is also an element of B. We use the notation A⊆B to indicate that A is a subset of the set B. If, instead, we want to stress that B is a superset of A, we use the

equivalent notation B⊇A. (So, A⊆B and B⊇A are equivalent statements.)

Let A and B be sets. The difference of A and B, denoted by A−B, is the set containing those elements that are in A but not in B. The difference of A and B is also called the

complement of B with respect to A

Let A and B be sets. The intersection of the sets A and B, denoted by A∩B, is the set

containing those elements in both A and B.

The intersection of a collection of sets is the set that

contains those elements that are members of all the sets in the collection

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.

⌊ x ⌋ is which function?

floor function

Once we have proved set identities, we can use them to prove new identities. In particular, we can apply a

string of identities, one in each step, to take us from one side of a desired identity to the other.

One way to show that two sets are equal is to show that each is a

subset of the other

One way is to list all the members of a set, when this is possible. We use a notation where all members of the set are listed between braces. For example, the notation {a,b,c,d}{a,b,c,d} represents the set with the four elements a, b, c, and d. This way of describing a set is known as

the roster method

The Ceiling Function is

the smallest integer greater than or equal to x

A set is an unordered collection of distinct objects, called elements or members of the set. A set is said

to contain its elements. We write a∈AA to denote that a is an element of the set A. The notation a∉A denotes that a is not an element of the set A.

Let S be a set. If there are exactly n distinct elements in S where n is a nonnegative integer, we say that S is a finite set and that n is the cardinality of S. The cardinality of S is denoted by

| S |