Discrete Mathematics and Its Applications ( 8th International Edition ) ISBN:9781260091991

8th Edition

ISBN: 9781259676512

Author: Kenneth H Rosen

Publisher: McGraw-Hill Education

See similar textbooks

Videos

Textbook Question

Chapter 2.6, Problem 1E

Let A = [ 1 1 1 3 2 0 4 6 1 1 3 7 ] .

What size is A?
What is the third column of A?
What is the second row of A?
What is the element of A in the (3, 2)th position?

A t

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 2.5, Problem 41E

Chapter 2.6, Problem 2E

Students have asked these similar questions

The everything combined problem Suppose that a computer science laboratory has 15 workstations and 10 servers. A cable can be used to directly connect a workstation to a server. For each server, only one direct connection to that server can be active at any time. 1. How many cables would you need to connect each station to each server? 2. How many stations can be used at one time? 3. How many stations can not be used at any one time? 4. How many ways are there to pick 10 stations out of 15? 5. (This one is tricky) We want to guarantee that at any time any set of 10 or fewer workstations can simultaneously access different servers via direct connections. What is the minimum number of direct connections needed to achieve this goal?

Can you help me with D and E

Q1.1 1 Point Any set {V1, V2, V3, V4} that consists of four different vectors from R cannot possibly span Rº. True False Save Answer

Chapter 2 Solutions

Discrete Mathematics and Its Applications ( 8th International Edition ) ISBN:9781260091991

Ch. 2.1 - List the members of these sets. { xx is a real...Ch. 2.1 - Use set builder notation to give a description of...Ch. 2.1 - Which of the intervals (0, 5), (0, 5], [0, 5), [0,...Ch. 2.1 - For each of these intervals, list all its elements...Ch. 2.1 - For each of these pairs of sets, determine whether...Ch. 2.1 - For each of these pairs of sets, determine whether...Ch. 2.1 - Prob. 7E Ch. 2.1 - Prob. 8E Ch. 2.1 - For each of the following sets, determine whether...Ch. 2.1 - Prob. 10E

Ch. 2.1 - Determine whether each of these statements is true...Ch. 2.1 - Determine whether these statements are true or...Ch. 2.1 - Determine whether each of these statements is true...Ch. 2.1 - Prob. 14E Ch. 2.1 - Use a Venn diagram to illustrate the set of all...Ch. 2.1 - Prob. 16E Ch. 2.1 - Use a Venn diagram to illustrate the re1ationships...Ch. 2.1 - Use a Venn diagram to illustrate the relationships...Ch. 2.1 - Prob. 19E Ch. 2.1 - Prob. 20E Ch. 2.1 - What is the cardinality of each of these sets? {a}...Ch. 2.1 - What is the cardinality of each of these sets? {}...Ch. 2.1 - Prob. 23E Ch. 2.1 - Prob. 24E Ch. 2.1 - How many elements does each of these sets have...Ch. 2.1 - Determine whether each of these sets is the power...Ch. 2.1 - Prove that P(A)P(B) if and only if AB .Ch. 2.1 - Show that if AC and BD , then ABCD Ch. 2.1 - Let A={a,b,c,d} and B={y,z} . Find AB . BA .Ch. 2.1 - Prob. 30E Ch. 2.1 - That is the Cartesian product ABC , where A is the...Ch. 2.1 - Prob. 32E Ch. 2.1 - Prob. 33E Ch. 2.1 - Let A={a,b,c} , B={x,y} , and C={0,l} . Find ABC ....Ch. 2.1 - Find A2 if A={0,1,3} A={1,2,a,b}Ch. 2.1 - Find A3 if A={a} A={0,a}Ch. 2.1 - How many different elements does AB have if A has...Ch. 2.1 - How many different elements does ABC have if A has...Ch. 2.1 - How many different elements does An have when A...Ch. 2.1 - Show that ABBA , when A and B are nonempty, unless...Ch. 2.1 - Explain why ABC and (AB)C are not the same.Ch. 2.1 - Explain why (AB)(CD) and A(BC)D are not the same.Ch. 2.1 - Prove or disprove that if A and B are sets, then...Ch. 2.1 - Prove or disprove that if A, B, and C are nonempty...Ch. 2.1 - Translate each of these quantifications into...Ch. 2.1 - Translate each of these quantifications into...Ch. 2.1 - Find the truth set of each of these predicates...Ch. 2.1 - Find the truth set of each of these predicates...Ch. 2.1 - Prob. 49E Ch. 2.1 - Prob. 50E Ch. 2.1 - Prob. 51E Ch. 2.2 - Prob. 1E Ch. 2.2 - Suppose that A is the set of sophomores at your...Ch. 2.2 - Let A={1,2,3,4,5} and B={0,3,6} . Find AB . AB ....Ch. 2.2 - Let A={a,b,c,d,e} and B={a,b,c,d,e,f,g,h} . Find...Ch. 2.2 - Prob. 5E Ch. 2.2 - Prob. 6E Ch. 2.2 - Prob. 7E Ch. 2.2 - Prob. 8E Ch. 2.2 - Prob. 9E Ch. 2.2 - Prob. 10E Ch. 2.2 - Prob. 11E Ch. 2.2 - Prob. 12E Ch. 2.2 - TABLE 1 Set Identities. Identity Name AU=AA=A...Ch. 2.2 - Prob. 14E Ch. 2.2 - Prob. 15E Ch. 2.2 - Prob. 16E Ch. 2.2 - Show that if A and B are sets in a universe U then...Ch. 2.2 - Prob. 18E Ch. 2.2 - Prob. 19E Ch. 2.2 - Prob. 20E Ch. 2.2 - Prob. 21E Ch. 2.2 - Prob. 22E Ch. 2.2 - Prob. 23E Ch. 2.2 - Prob. 24E Ch. 2.2 - Prob. 25E Ch. 2.2 - Let A, B, and C be sets. Show that (AB)C=(AC)(BC)...Ch. 2.2 - Prob. 27E Ch. 2.2 - Prob. 28E Ch. 2.2 - Prob. 29E Ch. 2.2 - Prob. 30E Ch. 2.2 - Prob. 31E Ch. 2.2 - Prob. 32E Ch. 2.2 - Let A and B be subsets of a universal set U. Show...Ch. 2.2 - Let A, B, and C be sets. Use the identity AB=AB ,...Ch. 2.2 - Prob. 35E Ch. 2.2 - Prob. 36E Ch. 2.2 - Prove or disprove that for all sets A, B, and C,...Ch. 2.2 - Prob. 38E Ch. 2.2 - Prob. 39E Ch. 2.2 - Prob. 40E Ch. 2.2 - Prob. 41E Ch. 2.2 - Prob. 42E Ch. 2.2 - The symmetric difference of A and B, denoted by AB...Ch. 2.2 - Prob. 44E Ch. 2.2 - Prob. 45E Ch. 2.2 - The symmetric difference of A and B, denoted by AB...Ch. 2.2 - Prob. 47E Ch. 2.2 - Prob. 48E Ch. 2.2 - Prob. 49E Ch. 2.2 - Prob. 50E Ch. 2.2 - The symmetric difference of A and B, denoted by AB...Ch. 2.2 - Prob. 52E Ch. 2.2 - The symmetric difference of A and B, denoted by AB...Ch. 2.2 - Prob. 54E Ch. 2.2 - The symmetric difference of A and B, denoted by AB...Ch. 2.2 - The symmetric difference of A and B, denoted by AB...Ch. 2.2 - The symmetric difference of A and B, denoted by AB...Ch. 2.2 - Prob. 58E Ch. 2.2 - The symmetric difference of A and B, denoted by AB...Ch. 2.2 - The symmetric difference of A and B, denoted by AB...Ch. 2.2 - The symmetric difference of A and B, denoted by AB...Ch. 2.2 - The symmetric difference of A and B, denoted by AB...Ch. 2.2 - The symmetric difference of A and B, denoted by AB...Ch. 2.2 - Prob. 64E Ch. 2.2 - Prob. 65E Ch. 2.2 - Prob. 66E Ch. 2.2 - Prob. 67E Ch. 2.2 - Prob. 68E Ch. 2.2 - Prob. 69E Ch. 2.2 - The successor of the set A is the set A{A} ....Ch. 2.2 - The Jaccard similarity J(A,B) of the finite sets A...Ch. 2.2 - Prob. 72E Ch. 2.2 - Prob. 73E Ch. 2.2 - Prob. 74E Ch. 2.2 - Prob. 75E Ch. 2.3 - Why is f not a function from R to R if f(x)=1/x?...Ch. 2.3 - Determine whether f is a function from Z to R if...Ch. 2.3 - Prob. 3E Ch. 2.3 - Find the domain and range of these functions. Note...Ch. 2.3 - Find the domain and range of these functions. Note...Ch. 2.3 - Find the domain and range of these functions. the...Ch. 2.3 - Find the domain and range of these functions. the...Ch. 2.3 - Find these values. 1.1 1.1 0.1 0.1 2.99 2.99 12+12...Ch. 2.3 - Find these values. 34 78 34 78 3 1 12+32 1252 Ch. 2.3 - Prob. 10E Ch. 2.3 - Which functions in Exercise 10 are onto? Determine...Ch. 2.3 - Determine whether each of these functions from Z...Ch. 2.3 - Prob. 13E Ch. 2.3 - Determine whether f:ZZZ is onto if f(m,n)=2mn ....Ch. 2.3 - Determine whether the function f:ZZZ is onto if...Ch. 2.3 - Consider these functions from the set of students...Ch. 2.3 - Consider these functions from the set of teachers...Ch. 2.3 - Specify a codomain for each of the functions in...Ch. 2.3 - Specify a codomain for each of the functions in...Ch. 2.3 - Prob. 20E Ch. 2.3 - Give an explicit formula for a function from the...Ch. 2.3 - Determine whether each of these functions is a...Ch. 2.3 - Determine whether each of these functions is a...Ch. 2.3 - Let f:RR and let f(x)0 for all xR . Show that f(x)...Ch. 2.3 - Let f:RR and 1et f(x)0 for all xR . Show that f(x)...Ch. 2.3 - Prove that a strictly increasing function from R...Ch. 2.3 - Prob. 27E Ch. 2.3 - Show that the function f(x)=ex from the set of...Ch. 2.3 - Prob. 29E Ch. 2.3 - Let S={1,0,2,4,7} . Find f(S) if f(x)=1 ....Ch. 2.3 - Let f(x)=x2/3 . Find f(S) if S={2,1,0,1,2,3}...Ch. 2.3 - Let f(x)=2x where the domain is the set of real...Ch. 2.3 - Prob. 33E Ch. 2.3 - Suppose that g is a function from A to B and f is...Ch. 2.3 - Prob. 35E Ch. 2.3 - If f and fog are one-to-one, does it follow that g...Ch. 2.3 - Prob. 37E Ch. 2.3 - Find fog and gof where f(x)=x2 and g(x)=x+2 , are...Ch. 2.3 - Prob. 39E Ch. 2.3 - Let f(x)ax+b and g(x)=cx+d , where a, b, c, and d...Ch. 2.3 - Show that the function f(x)ax+b from R to R, where...Ch. 2.3 - Prob. 42E Ch. 2.3 - Prob. 43E Ch. 2.3 - Let f be the function from R to R defined by...Ch. 2.3 - Let g(x)=|x| . Find g1({0}) . g1({1,0,1}) ....Ch. 2.3 - Prob. 46E Ch. 2.3 - Prob. 47E Ch. 2.3 - Show x+12 is the closest integer to the number x...Ch. 2.3 - Prob. 49E Ch. 2.3 - Show that if x is a real number, then xx=1 if x is...Ch. 2.3 - Prob. 51E Ch. 2.3 - Prob. 52E Ch. 2.3 - Prob. 53E Ch. 2.3 - Show that if x is a real number and n is an...Ch. 2.3 - Prob. 55E Ch. 2.3 - Prove that if x is a real number, then x=x and x=x...Ch. 2.3 - Prob. 57E Ch. 2.3 - Prob. 58E Ch. 2.3 - Prob. 59E Ch. 2.3 - How many bytes are required to encode n bits of...Ch. 2.3 - How many bytes are required to encode n bits of...Ch. 2.3 - How many ATM cells (described in Example 30) can...Ch. 2.3 - Data are transmitted over a particular Ethernet...Ch. 2.3 - Draw the graph of the function f(n)=1n2 from Z to...Ch. 2.3 - Draw the graph of the function f(x)=2x from R to...Ch. 2.3 - Draw the graph of the function f(x)=x/2 from R to...Ch. 2.3 - Prob. 67E Ch. 2.3 - Draw the graph of the function f(x)=x+x/2 from R...Ch. 2.3 - Draw graphs of each of these functions. f(x)=x+12...Ch. 2.3 - Prob. 70E Ch. 2.3 - Find the inverse function of f(x)=x3+1 .Ch. 2.3 - Suppose that f is an invertible function from Y to...Ch. 2.3 - Let S be a subset of a universal set U. The...Ch. 2.3 - Suppose that f is a function from A to B, where A...Ch. 2.3 - Prove or disprove each of these statements about...Ch. 2.3 - Prove or disprove each of these statements about...Ch. 2.3 - Prove that if x is a positive real number, then...Ch. 2.3 - Let x be a real number. Show that 3x=x+x+13+x+23 .Ch. 2.3 - For each of these partial functions, determine its...Ch. 2.3 - Prob. 80E Ch. 2.3 - Prob. 81E Ch. 2.3 - Show that a set S is infinite if and only if there...Ch. 2.4 - Find these terms of the sequence {an} , where...Ch. 2.4 - What is the term a8 of the sequence {an} if an ,...Ch. 2.4 - What are the terms a0,a1,a2 , and a3 of the...Ch. 2.4 - What are the terms a0,a1,a2 , and a3 of the...Ch. 2.4 - List the first 10 terms of each of these...Ch. 2.4 - List the first lo terms of each of these...Ch. 2.4 - Find at least three different sequences beginning...Ch. 2.4 - Find at least three different sequences beginning...Ch. 2.4 - Find the first five terms of the sequence defined...Ch. 2.4 - Find the first six terms of the sequence defined...Ch. 2.4 - Let an=2n+53n for n=0,1,2,,... Find a0,a1,a2,a3 ,...Ch. 2.4 - Show that the sequence {an} is a solution of the...Ch. 2.4 - Is the sequence {an} a solution of the recurrence...Ch. 2.4 - For each of these sequences find a recurrence...Ch. 2.4 - Show that the sequence {an} is a solution of the...Ch. 2.4 - Find the solution to each of these recurrence...Ch. 2.4 - Find the solution to each of these recurrence...Ch. 2.4 - A person deposits $1000 in an account that yields...Ch. 2.4 - Suppose that the number of bacteria in a colony...Ch. 2.4 - Assume that the population of the world in 2017...Ch. 2.4 - A factory makes custom sports cars at an...Ch. 2.4 - An employee joined a company in 2017 with a...Ch. 2.4 - Find a recurrence relation for the balance B(k)...Ch. 2.4 - Find a recurrence relation for the balance B(k)...Ch. 2.4 - For each of these lists of integers, provide a...Ch. 2.4 - For each of these lists of integers, provide a...Ch. 2.4 - *27. Show that if an denotes the nth positive...Ch. 2.4 - Let an , be the nth term of the sequence 1, 2, 2,...Ch. 2.4 - What are the values of these sums? k=15(k+1)...Ch. 2.4 - What are the values of these sums, where...Ch. 2.4 - What is the value of each of these sums of terms...Ch. 2.4 - Find the value of each of these sums. j=08(1+ ( 1...Ch. 2.4 - Compute each of these double sums. i=12j=13( i+j)...Ch. 2.4 - Compute each of these double sums. i=13j=12( i+j)...Ch. 2.4 - Show that j=1n(aja j1)=ana0 , where a0,a1,...,an...Ch. 2.4 - Use the identity 1/(k(k+1))=1/k1/(k+1) and...Ch. 2.4 - Sum both sides of the identity k2(k21)2=2k1 from...Ch. 2.4 - Use the technique given in Exercise 35, together...Ch. 2.4 - Find k=100200k . (Use Table 2.) TABLE 2 Some...Ch. 2.4 - Prob. 40E Ch. 2.4 - Find k=1020k2(k3) . (Use Table 2.) TABLE 2 Some...Ch. 2.4 - Find . k=1020(k1)(2k2+1) (Use Table 2.) TABLE 2...Ch. 2.4 - Find a formula for k=0mk , when m is a positive...Ch. 2.4 - Find a formula for k=0mk3 , when m is a positive...Ch. 2.4 - There is also a special notation for products. The...Ch. 2.4 - Express n! using product notation.Ch. 2.4 - Find j=04j! .Ch. 2.4 - Find j=04j! .Ch. 2.5 - Prob. 1E Ch. 2.5 - Determine whether each of these sets is finite,...Ch. 2.5 - Determine whether each of these sets is countable...Ch. 2.5 - Determine whether each of these sets is countable...Ch. 2.5 - Show that a finite group of guests arriving at...Ch. 2.5 - Suppose that Hilbert’s Grand Hotel is fully...Ch. 2.5 - Suppose that Hilbert’s Grand Hotel is fully...Ch. 2.5 - Show that a countably infinite number of guests...Ch. 2.5 - Suppose that a countably infinite number of buses,...Ch. 2.5 - Give an example of two uncountable sets A and B...Ch. 2.5 - Give an example of two uncountable sets A and B...Ch. 2.5 - Prob. 12E Ch. 2.5 - Prob. 13E Ch. 2.5 - Prob. 14E Ch. 2.5 - Prob. 15E Ch. 2.5 - Show that a subset of a countable set is also...Ch. 2.5 - Prob. 17E Ch. 2.5 - Prob. 18E Ch. 2.5 - Prob. 19E Ch. 2.5 - Show that if |A|=|B| and |B|=|C| , then |A|=|C| .Ch. 2.5 - Prob. 21E Ch. 2.5 - Suppose that A is a countable set. Show that the...Ch. 2.5 - Prob. 23E Ch. 2.5 - Prob. 24E Ch. 2.5 - Prob. 25E Ch. 2.5 - Prob. 26E Ch. 2.5 - Show that the union of a countable number of...Ch. 2.5 - Show that the set Z+Z+ is countable Ch. 2.5 - Prob. 29E Ch. 2.5 - Show that the set of real numbers that are...Ch. 2.5 - Show that Z+Z+ t is countable by showing that the...Ch. 2.5 - Show that when you substitute (3n+1)2 for each...Ch. 2.5 - Prob. 33E Ch. 2.5 - Show that (0, 1) and R have the same cardinality...Ch. 2.5 - Prob. 35E Ch. 2.5 - Prob. 36E Ch. 2.5 - Show that the set of all computer programs in a...Ch. 2.5 - Prob. 38E Ch. 2.5 - Prob. 39E Ch. 2.5 - Show that if S is a set, then there does not exist...Ch. 2.5 - In this exercise, we prove the Schröder-Bernstein...Ch. 2.6 - Let A=[111320461137] . What size is A? What is the...Ch. 2.6 - Find A + B, where A=[104122022],B=[135223230]...Ch. 2.6 - Find AB if A=[2132],B=[0413] A=[110123],B=[321102]...Ch. 2.6 - Find the product AB, where...Ch. 2.6 - Find a matrix A such that [2314]A=[3012] . [Hint:...Ch. 2.6 - Find a matric A such that [132211403]A=[713103137]Ch. 2.6 - Prob. 7E Ch. 2.6 - Prob. 8E Ch. 2.6 - Prob. 9E Ch. 2.6 - Prob. 10E Ch. 2.6 - Prob. 11E Ch. 2.6 - In this exercise we show that matrix...Ch. 2.6 - Prob. 13E Ch. 2.6 - The nn matrix A=[aij] is called a diagonal matrix...Ch. 2.6 - Let A=[1101] . Find a formula for An , whenever n...Ch. 2.6 - Show that (At)t=A .Ch. 2.6 - Prob. 17E Ch. 2.6 - Show that [231121113] Is the inverse of...Ch. 2.6 - Prob. 19E Ch. 2.6 - Prob. 20E Ch. 2.6 - Prob. 21E Ch. 2.6 - Prob. 22E Ch. 2.6 - Prob. 23E Ch. 2.6 - Prob. 24E Ch. 2.6 - Prob. 25E Ch. 2.6 - Let A=[1101] and B=[0110] Find AB . AB . AB .Ch. 2.6 - Prob. 27E Ch. 2.6 - Find the Boolean product of A and B, where...Ch. 2.6 - Prob. 29E Ch. 2.6 - Let A be a zeroone matrix. Show that AA=A . AA=A .Ch. 2.6 - Prob. 31E Ch. 2.6 - Prob. 32E Ch. 2.6 - Prob. 33E Ch. 2.6 - Prob. 34E Ch. 2.6 - In this exercise we will show that the Boolean...Ch. 2 - Prob. 1RQ Ch. 2 - What is the empty set? Show that the empty set is...Ch. 2 - Define |S|, the cardinality of the set S. Give a...Ch. 2 - Define the power set of a set S. When is the empty...Ch. 2 - Define the union. intersection, difference, and...Ch. 2 - Prob. 6RQ Ch. 2 - Explain the relationship between logical...Ch. 2 - Prob. 8RQ Ch. 2 - Prob. 9RQ Ch. 2 - Define the inverse of a function. When does a...Ch. 2 - Prob. 11RQ Ch. 2 - Conjecture a formula for the terms of the sequence...Ch. 2 - Prob. 13RQ Ch. 2 - What is the sum of the terms of the geometric...Ch. 2 - Show that the set of odd integers is countable.Ch. 2 - Prob. 16RQ Ch. 2 - Prob. 17RQ Ch. 2 - Prob. 18RQ Ch. 2 - Prob. 1SE Ch. 2 - Prob. 2SE Ch. 2 - Prob. 3SE Ch. 2 - Prob. 4SE Ch. 2 - Prob. 5SE Ch. 2 - Prob. 6SE Ch. 2 - Prob. 7SE Ch. 2 - Prob. 8SE Ch. 2 - Prob. 9SE Ch. 2 - Prob. 10SE Ch. 2 - Prob. 11SE Ch. 2 - Prob. 12SE Ch. 2 - Prob. 13SE Ch. 2 - Prob. 14SE Ch. 2 - Prob. 15SE Ch. 2 - *16. Suppose that f is a function from the set A...Ch. 2 - Prob. 17SE Ch. 2 - Prob. 18SE Ch. 2 - Prob. 19SE Ch. 2 - Prob. 20SE Ch. 2 - Prob. 21SE Ch. 2 - Prob. 22SE Ch. 2 - Prob. 23SE Ch. 2 - Prove that if x is a real number, then x/2/2=x/4 .Ch. 2 - Prob. 25SE Ch. 2 - Prob. 26SE Ch. 2 - Prove that if m is a positive integer and x is a...Ch. 2 - We define the Ulam numbers by setting u1=1 and...Ch. 2 - Prob. 29SE Ch. 2 - Determine a rule for generating the terms of the...Ch. 2 - Prob. 31SE Ch. 2 - Prob. 32SE Ch. 2 - Prob. 33SE Ch. 2 - Show that the set of all finite subsets of the set...Ch. 2 - Prob. 35SE Ch. 2 - Prob. 36SE Ch. 2 - Prob. 37SE Ch. 2 - Prob. 38SE Ch. 2 - Prob. 39SE Ch. 2 - Prob. 40SE Ch. 2 - Prob. 41SE Ch. 2 - Prob. 1CP Ch. 2 - Prob. 2CP Ch. 2 - Prob. 3CP Ch. 2 - Prob. 4CP Ch. 2 - Prob. 5CP Ch. 2 - Prob. 6CP Ch. 2 - Prob. 7CP Ch. 2 - Prob. 8CP Ch. 2 - Prob. 9CP Ch. 2 - Prob. 10CP Ch. 2 - Prob. 11CP Ch. 2 - Prob. 12CP Ch. 2 - Prob. 1CAE Ch. 2 - Prob. 2CAE Ch. 2 - Use a computational program or programs you have...Ch. 2 - Prob. 4CAE Ch. 2 - Prob. 5CAE Ch. 2 - Use a computational program or programs you have...Ch. 2 - Prob. 1WP Ch. 2 - Research where the concept of a function first...Ch. 2 - Explain the different ways in which the...Ch. 2 - Define the recently invented EKG sequence and...Ch. 2 - Prob. 5WP Ch. 2 - Expand the discussion of the continuum hypothesis...

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

Let A = [ 1 1 1 3 2 0 4 6 1 1 3 7 ] . What size is A? What is the third column of A? What is the second row of A? What is the element of A in the (3, 2)th position? What is A t ?

Videos

Want to see the full answer?

Chapter 2 Solutions