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MATH 301 FINAL
| Question | Answer |
|---|---|
| One way to disprove a statement of the form "if A then B" is to | find an instance where A is true and B is false |
| Which of the following is the same as the (false) statement "The product of two negative integers in negative"? | If x and y are negative integers, then xy is negative |
| A bit string (or binary string) is a list of 0s and 1s. How many length-k bit strings are there? | 2 to the power of k |
| How many binary sequences of length 5 are there? | 32 |
| How many ternary sequences of length 5 are there? | 243 |
| How many ternary sequences of length 5 are there which use exactly one 2? | 80 |
| Given that the value of 9! is 362880, what is the value of 10! ? | 3628800 |
| Identify whether each of the following is true or false. (n)vn=n^2 | False |
| Identify whether the following is true or false. (n)vn=n! | True |
| Identify whether the following is true or false. n!=n(n-1)! | True |
| Identify whether the following is true or false. (n+1)!=(n+1)n! | True |
| Identify whether each of the following is true or false. n!-n^(n-1) | False |
| Calculate sum subscript k equals 1 end subscript superscript 3 space k squared | 14 |
| Calculate product subscript k equals 1 end subscript superscript 3 space k squared | 36 |
| Calculate product subscript k equals 1 end subscript superscript 3 space 1 over k | 1/6 |
| A PIN is a 4 digit number, which might start with a 0. How many different PINs do not have any repeated digits? | 5,040 |
| Let X= { {1,2} , {3,4,5} }. Is {1,2} an element of X? | Yes |
| Let X= { {1,2} , {3,4,5} }. Is {1} an element of X? | No |
| Let X= { {1,2} , {3,4,5} }. Is {1,2,3,4} an element of X? | No |
| Does {x∈N: 2|x and x<10} define the set B= {0,2,4,6,8}? | Yes |
| Does {x∈Z: 2|x and x<10} define the set B= {0,2,4,6,8}? | No |
| Does {2n+4 : n∈Z, -2≤n≤2} define the set B= {0,2,4,6,8}? | Yes |
| Does {2n: n∈Z, -2≤n≤2} define the set B= {0,2,4,6,8}? | No |
| Let A= {1,2,3,4,5}. Is {1} a subset of A? | Yes |
| Let A= {1,2,3,4,5}. Is {1,2,3,4} a subset of A? | Yes |
| Let A= {1,2,3,4,5}. Is {3,4,5,6} a subset of A? | No |
| Let A= {1,2,3,4,5}. Is {Ø} a subset of A? | Yes |
| Let A= {1,2,3,4,5}. Is {{1,2},{3,4,5}} a subset of A? | No |
| There is some integer... | ∃x∈Z |
| For every integer... | ∀x∈Z |
| There is some integer which is less than zero | ∃x∈Z, x<0 |
| Every integer is less than zero | ∀x∈Z,x<0 |
| There is some real number whose square is 3 | ∃a∈R, a^2=3 |
| we say b|a provided (divisibility) | ∃c∈Z, a=bc |
| There is some real number y so that for every non zero real number x, xy=1 (T/F) | False |
| Given any non-zero real number x, there is some real number y so that xy=1 | True |
| Let A = {1,2,3,4,5} and B={4,5,6,7}. Identify the elements of A union B | 1,2,3,4,5,6,7 |
| Let A = {1,2,3,4,5} and B={4,5,6,7}. Identify the elements of A intersection B | 4,5 |
| Let A = {1,2,3,4,5} and B={4,5,6,7}. Identify the elements of A minus B. | 1,2,3 |
| Let A = {1,2,3,4,5} and B={4,5,6,7}. Identify the elements of B minus A. | 6,7 |
| Let A = {1,2,3,4,5} and B={4,5,6,7}. Identify the elements of A ∆B. | 1,2,3,6,7 |
| A T-shirt shop offers a particular style in three different colors, long sleeve and short sleeve, and sizes S, M, L, and XL. How many different combinations of color, sleeve style, and size are possible? | 288 |
| A group of 10 students is attending a special presentation after which the speaker will take some questions. The group can select 4 students who will be able to ask questions. How many different ways are there to select and line up those 4 people? | 5,040 |
| A ternary sequence is made up of 0s, 1s, and 2s. How many ternary sequences of length 5 are there? | 243 |
| a shelf contains 100 different books.50 math, 30 physics, 20 history. Sam wants to select one book which is about either math or physics. How many different selections are possible? | 80 |
| 100 books; 50 math 30 physics 20 history. If Sam wanted to select two books, one on math and the other on physics, how many different selections would be possible | 1500 |
| Calculate (10)v3 | 720 |
| Calculate the following value: 1000!/998! | 999,000 |
| Identify whether the statement is true or false. 3|9 | True |
| Identify whether the statement is true or false. 9|3 | False |
| Identify whether the statement is true or false. 2|0 | True |
| Identify whether the statement is true or false. 0|2 | False |
| Identify whether the statement is true or false. 0|0 | True |
| Write using mathematical notation in if, then form. The sum of 2 odd integers is even. | if x and y are odd integers, then x + y is even |
| What is a counterexample? | Instance where A is true and B is false |
| There is some integer x such that the square of any integer is at least as large as x | ∃x∈Z, ∀y∈Z, y^2≥x |
| Write the statement in English and identify whether it is true or false (prove). ∀x∈Z, ∃y∈Z, y<x^3 | This says that for every integer x there is an integer y which is less than the cube of x. True. Given integer x, we can simply let y=x^3 -1 |
| What correctly defines the set of all integer factors of 12? | {x∈Z: x|12 and x≤12} |
| Let A= {x∈Z: x=4m for some int m} and let B= {x∈Z: x=4^k for some int k}. Prove or disprove that A⊆B. | This is false since 8=4(2) we have 8∈A, but 8 is not 4^k for any integer k, so 8 ∉B. |
| Let A= {x∈Z: x=4m for some int m} and let B= {x∈Z: x=4^k for some int k}. Prove or disprove that B⊆A. | This is false since 4^0 =1, so 1∈B, but 1 ∉ A. |
| Let A= {x∈Z: x=4m for some int m} and let B= {x∈Z: x=4^k for some int k}. Prove or disprove that A=B | This is false. Neither A⊆B or B⊆A. is true so they are not equal. |
| Suppose A= {1,2,3,4,5} and B= {2,4,6,8}. What is A ⋃ B? | {1,2,3,4,5,6,7,8,} |
| Suppose A= {1,2,3,4,5} and B= {2,4,6,8}. What is A⋂ B? | {2,4} |
| Suppose A= {1,2,3,4,5} and B= {2,4,6,8}. What is A- B? | {1,3,5} |
| Suppose A= {1,2,3,4,5} and B= {2,4,6,8}. What is B-A? | {6,8} |
| Suppose A= {1,2,3,4,5} and B= {2,4,6,8}. What is A ∆ B? | {1,3,5,6,8} |
| Let A= {1,2,3,4,5} and B= {x,y,z}. is { (1,x), (3,y), (5,x) } a relation from A to B? | Yes |
| Let A= {1,2,3,4,5} and B= {x,y,z}. is { (1,2), (3,4), (5,x), (y,z) } a relation from A to B? | No |
| Let A= {1,2,3,4,5} and B= {x,y,z}. is { (x,1), (y,2), (z,3) } a relation from A to B? | No |
| Let A = {1,2,3,4,5} and B = {x,y,z}. is { (1,2), (3,4), (5,x), (y,z) } a relation on A? | No |
| Let A= {1,2,3,4,5} and B= {x,y,z}. is { (1,2), (3,4), (5,x), (y,z) } a relation on A? | No |
| Let A= {1,2,3,4,5} and B= {x,y,z}. is { (x,1), (y,2), (z,3) } a relation on A? | No |
| Suppose that R is a relation on set A. For each x in A, x is related to x | R is reflexive |
| Suppose that R is a relation on set A. For each x in A, x is not related to x | R is irreflexive |
| Suppose that R is a relation on set A. If x is related to y, then y is related to x | R is symmetric |
| Suppose that R is a relation on set A. If x is related to y and y is related to x, then x=y | R is antisymmetric |
| Suppose that R is a relation on set A. If xRy and yRz, then xRz | R is transitive |
| Consider the relation R= {x,y} : x,y∈Z, |x-y|≤2} on the set of integers. Identify what of the five properties are satisfied by R. | R is reflexive on Z and symmetric. |
| Consider the relation R= {x,y} : x,y∈Z, xy≥0} on the set of integers. Identify what of the five properties are satisfied by R. | R is reflexive on Z and symmetric |
| Let A={1,2,3}. Consider the relation R={(1,1), (1,2), (1,3), (2,2), (2,3), (3,3)} on the set A. Identify what of the 5 properties are satisfied by R. | R is reflective on A, antisymmetric and transitive |
| Let R be the relation has-the-same-size-as on 2 ^{1,2,3,4,5}. Consider the equivalence class [{1,3}] for relation R. What is the class subset size? | Subsets of {1,2,3,4,5} have size 2 |
| Let A={1,2,3,4} and let R be the equivalence relation on 2^A with X R Y if and only if |X|=|Y|. How many equivalence classes does R have? | 5 |
| How many different ways are there to order (all five of) the letter AAAAC ? | 5 |
| How many different ways are there to order (all five of) the letter AAABC ? | 20 |
| How many different ways are there to order (all five of) the letter AAACC ? | 10 |
| Which of the following properties are required for the collection of sets Bv1, Bv2, ... Bvn to be a partition of A? | 1. Each Bvi to be nonempty, 2. The intersection of any 2 sets in a Bv1,Bv2,...Bvn is empty, 3 Union of the sets is A, and each Bvi is a subset of A |
| Where n and k are non-negative integers, the symbol (n k) denotes | The number of k-element subsets of an n-element set |
| (n 0)= | 1 |
| (n 1)= | n |
| (n n-1)= | n |
| (n n)= | 1 |
| The first 6 rows of Pascal's triangle are shown below. 1 1,1 1,2,1 1,3,3,1 1,4,6,4,1 1,5,10,10,5,1 Calculate the next row | 1,6,15,20,15,6,1 |
| Use the binomial theorem to find the numeric coefficient of x^2y in the expansion of (3x+9y)^3 | The coefficient is 243 |
| Use binomial theorem to simplify, ∑(197 k)(-1)^k | 0 |
| How many distinct 5-digit sequences are there, if sequences which are reverse orderings of each other are considered the same? For example, 12345 is the same as 54321, and 11230 is the same as 03211. | 50,500 |
| n>20. Pick 3 (P, VP S) from 20. Then, pick 5 out of the rest. How would this be done with the multiplication principle? | (n 20) (20)v3 (17 5) |
| n>20. Pick 3 (P, VP S) from 20. Then, pick 5 out of the rest. What would it look like to go down in a line (first in line P, second VP, third S) then the next 5 select, and rest in line are selected. | n!/(5! 12! (n-20)! |
| This class has 46 students. I have 12 exercises which I would like to assign. Number of different ways to assign one exercise to each student. | 12^46 |
| This class has 46 students. I have 12 exercises which I would like to assign. Number of different ways to select one student to present each exercise. | (46)v12 |
| This class has 46 students. I have 12 exercises which I would like to assign. Number of different ways to select 12 students, each of whom does all of the exercises | (46 12) |
| This class has 46 students. I have 12 exercises which I would like to assign. Number of different ways to create 46 worksheets, each of which contains just 1 exercise. | (( 12 46)) |
| In how many different ways could we distribute 10 identical candies among 4 children? ( Do allow the possibility of some children not getting any) | 286 |
| How many ways are there to choose a collection of eight coins from a piggy bank containing 100 identical pennies, 100 identical nickels, and 100 identical dimes? | 45 |
| There are three large groups of people, each with 1000 members. Any two of these groups have 100 members in common. There are 10 people who are in all three groups. All together, how many people are in these groups? | 2,710 |
| Two integers are called relatively prime if their largest common divisor is 1. Suppose we want to show that some integer n is relatively prime to 30.What would be the smallest set of conditions which must be shown | n is not divisible by 2, 3, 5 |
| Complete the following statement of the binomial theorem. You will need to use the math palette to enter your answer. Let n∈N. Then (x+y)^n= | ∑(n k) x^(n-k)y^k |
| Use binomial theorem to simplify the following sum as much as possible ∑(-1)^k(2018 k) 2018 | (1-1)^2018=0^2018=0 |
| Use the binomial theorem to find the coefficient of x^3 in the expansion of (1+2x)^8. Simplify answer. | 448 |
| Simplify (7 2) | 21 |
| Simplify ((7 2)) | 28 |
| Find the number of distinct permutations of the letters in ENGINEER | 3,360 |
| Let R be the relation defined as follows. R={(x,y): x∈Z, y∈Z, |x|≤|y|} is this relation reflexive? Justify | Yes. Both x and y hold ability to match each other and be less than or equal to each other |
| Let R be the relation defined as follows. R={(x,y): x∈Z, y∈Z, |x|≤|y|} Is this relation symmetric? | No. Because x is always less than or equal to y, so they can't switch |
| Let R be the relation defined as follows. R={(x,y): x∈Z, y∈Z, |x|≤|y|} Is this relation transitive? Justify. | Yes. if y is less than or equal to z, then that means that x would be less than or equal to z |
| In how many different ways can 12 identical candy bars be distributed among 4 children? | 455 |
| Identify the equivalence classes for the following equivalence relation R on the set {a,b,c,d,e}. (You do not need to prove that it is an equivalence relation.) R = { (a,a), (b,b), (c,c), (d,d), (e,e), (a,b), (a,e), (b,a), (b,e), (e,a), (e,b) } | {a,b,e},{c},{d} |
| Test on pink, yellow, green paper. 28 students. How many ways to...Distribute the tests so that each student gets one. We are concerned only with which color form they get. | 3^28 |
| Test on pink, yellow, green paper. 28 students. How many ways to...Distribute exactly 10 of the pink form, 10 of the yellow, and 8 of the green. | 28!/(10!10!8!) |
| Test on pink, yellow, green paper. 28 students. How many ways to... Select a supply of the pink, yellow, and green forms so that the total number of forms is 28? | (30 2) |
| 30 in math club, 25 in robotics, 32 esports. 8 in both math and robotics, 12 in math and esports, 6 in robotics and esports. 1 in all 3. How many people in total? | 62 |
| Negation of "x is red and even" | x is not red or x is not even |
| Find the contrapositive of If x is a floop, then x^2 is blue | if x^2 is not blue, then x is not a floop. |
| Contrapositive of if P is prime, then 2^p -2 is prime | if 2^p -2 is not prime, then p is not prime |
| is {(x,y): x,y∈Z, y=x^2} a function? | Yes |
| is {(x,y): x,y∈Z, y^2=x} a function? | No |
| is {(1,2),(2,3),(3,2),(4,1)} a function? | Yes |
| is {(1,2),(2,3),(3,2),(3,1)} a function? | No |
| Determine whether the following function is one-to-one (injective) and whether it is onto (surjective). f: Z-> Z defined by f(x)= x^2 | neither one-to-one nor onto |
| Determine whether the following function is one-to-one (injective) and whether it is onto (surjective). g: Z->z defined by g(x) = 2x | g in one-to one (injective) |
| Let E= x ∈N: 2|x}, the even non-negative integers and O={x∈N:∃c∈Z, x=2c+1), the odd non-negative integers. determine whether the function g: O->Z defined by g(x) =(n+1)/2 is one-to-one and whether it is onto | g is one-to-one |
| Let E= x ∈N: 2|x}, the even non-negative integers and O={x∈N:∃c∈Z, x=2c+1), the odd non-negative integers. Determine whether the function f: E->Z defined by f(x)= -n/2 is one-to-one and whether it is onto | f is one-to-one |
| Let E= x ∈N: 2|x}, the even non-negative integers and O={x∈N:∃c∈Z, x=2c+1), the odd non-negative integers. Determine whether the piecewise function F: N->Z defined below is one-to-one, and whether it is onto. f(x) { -n/2 (if n is even), (n+1)/2 (odd n) | F is one-to-one |
| Suppose G=(V,E) is a graph if {a,b}∈E, we say that vertex a is.. | incident |
| Suppose G=(V,E) is a graph if {a,b}∈E, we say that vertex a is incident with edge {a,b}, and the vertices a and b are | adjacent |
| Suppose G=(V,E) is a graph if {a,b}∈E, we say that vertex a is incident with edge {a,b}, and the vertices a and b are adjacent. We could also say a and b are... | neighbors |
| Suppose G=(V,E) is a graph. The number of a vertex v is called the .... of v | degree |
| 20 people attend a party. If each of these people shakes every other person's hand exactly once, how many handshakes take place? | 190 |
| Suppose we form a graph where the vertices represent the twenty people at the party mentioned above, and two vertices are adjacent if those two people shake hands. How many edges does this graph have? | 190 |
| Suppose graph G has 12 vertices, and each vertex has degree 4. How many edges does G have? | 24 |
| Let G be the graph with vertex set V = {u,v,w,x,y} and edge set E = {uv, uw, ux, uy, vw, wx}Let H1=(V1, E1) V1 = {u,v,w,x}, E1 = {uv, uw, ux}. What is H1? | A subgraph of G, which is neither induced not spanning |
| Let G be the graph with vertex set V = {u,v,w,x,y} and edge set E = {uv, uw, ux, uy, vw, wx}. H2= (V2, E2), V2 = {u,v,w,x,y}, E2 = {vw, wx, uy} what is H2? | A spanning subgraph of G |
| Let G be the graph with vertex set V = {u,v,w,x,y} and edge set E = {uv, uw, ux, uy, vw, wx}, H3=(V3,E3), V3 = {u,v,w}, E3 = {uv, uw, vw}.What is H3? | An induced subgraph of G |
| Let G be the graph with vertex set V = {u,v,w,x,y} and edge set E = {uv, uw, ux, uy, vw, wx}, H4 = (V4,E4), V4 = {v,w,x}, E4 = {vw, vx, wx}, what is H4? | Not a subgraph of G |
| Graph G is called connected provided... | every pair of vertices u and v in G, there is a u-v path in G and G has at most 1 component |
| What 4 properties are true for any tree T? | 1. Connected 2. No Cycles 3. there is a unique a-b path for any 2 vertices a&b 4. graph T-e is disconnected for any edge e |
| Suppose that G is a subgraph of H. can a(G)< a(H) be true? | Yes |
| Suppose that G is a subgraph of H. Can w(G)>w(H) be true? | No |
| Suppose that G is a subgraph of H. can w(G)<w(H) be true? | Yes |
| Suppose that G is a subgraph of H. is w(G)≤w(H) true? | Yes and MUST be true. |
| The wheel graph W5 has vertex set V = {a1, a2, a3, a4, a5, b}, with a1 adjacent to a5, each ai adjacent to ai+1, and b adjacent to every ai. What is the chromatic number of this graph? | 4 |
| The notation Cn refers to a cycle graph with n vertices. The chromatic number of C16 is | 2 |
| The notation Cn refers to a cycle graph with n vertices. The clique number of C16 is | 2 |
| The notation Cn refers to a cycle graph with n vertices. The size of the largest possible independent set in C16 is | 8 |
| The notation Cn refers to a cycle graph with n vertices. The chromatic number of C23 is | 3 |
| The notation Cn refers to a cycle graph with n vertices. The clique number of C23 is | 2 |
| The notation Cn refers to a cycle graph with n vertices. The size of the largest possible independent set in C23 is | 11 |
| How many injective functions from a 4-element set to a 7-element set? | 840 |
| How many edges in a tree with 25 vertices? | 24 |
| Does a graph with a degree sequence 2,2,2,4,4,4 have a Euler trail? | Yes |
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