Section 1.1 solutions Homework #1 – 8. List all of the elements in the sets described below. Write your answer using set braces. 1) The set A of even numbers between 1 and 10 inclusive. My answer can only include numbers between 1 and 10. It is okay to include the endpoints because of the word inclusive. However, 1 won’t be in my answer as 1 is not an even number. Here are the even numbers between 1 and 10 are: 2,4,6,8,10 10 is an end point and it is even. 10 will be part of my answer. Answer: {2,4,6,8,10} 3) The set B of even numbers between 1 and 10 exclusive. My answer can only include numbers between 1 and 10. It is NOT okay to include the endpoints of 1 and 10 because of the word exclusive. Here are the even numbers between 1 and 10 are: 2,4,6,8,10 10 is an end point and it can’t be part of my answer. 2 is not an end point, so it is okay to put 2 in my answer. Answer: {2,4,6,8}

5) The set A of solutions to the problem: x + 6 = 10 I need to solve the problem for x: x + 6 = 10 ___-6__-6 x=4 The answer to this is the only number that goes in my set braces. Answer: {4}

7) The set B of flavors of ice cream in Neapolitan ice cream. This may take some research if you don’t already know the answer. Answer: {chocolate, vanilla, strawberry} Homework #9 – 16. Which of the following sets are well-defined. Write yes for your answer if the set is well-defined. Write no for your answer if the set is not well-defined. 9) A is the set of goofy dogs. Answer: NO not well defined, whether a dog is goofy or not is an opinion and different people may classify dogs differently. 11) D is the set of numbers whose square is 16. Answer: Yes I can create an equation and use Algebra to find the elements of the set. The elements are 4 and -4. 13) The set of years that the Arizona Cardinals won their division. Answer: Yes 15) The set of cities that have nice climates. Answer: No Nice climate is a matter of opinion. Some people will say Phoenix has a nice climate and others will say it does not. This is the case with most cities. Homework #17 – 24. Determine whether each set is finite or infinite 17) The set of factors of 12. The factors of 12 are: 1,2,3,4,6,12 There are only 6 factors of 12. Answer: Finite

19) The set of multiples of 3. These are the multiples of 3: {3,6,9,12,15,18…} It is not possible to list all of the multiples of 3. Answer: infinite 21) The set of multiples of 4 that are less than 30. The only multiples of 4 that are less than 30 are 4,8,12,16,20,24,28 It is possible to list them all. Answer: Finite 23) The set of even natural numbers. This consists of: {2,4,6,8,10…} It is impossible to list all of the members of this set.

Answer: Infinite Homework #25 – 34. Find the cardinal number of the following sets. That is find the number of elements in each set. 25) The set B of natural numbers between 5 and 10 inclusive. B contains the elements: 5,6,7,8,9,10 B has 6 elements Answer: n(B) = 6 27) The set A of even natural numbers greater than 4 and less than 11. A contains the elements: 6,8,10 A has 3 elements Answer: n(A) = 3

29) The set A of natural numbers less than 9. A contains the elements: 1,2,3,4,5,6,7,8 A has 8 elements Answer: n(A) = 8 31) Find n(A) where A = {1,3,5} A has 3 elements. Answer: n(A) = 3 33) Find n(C) where C = {2,4,6,8,10} C has 5 elements. Answer: n(C) = 5

Homework #35– 45. Determine whether the sets A and B are equal, equivalent, neither. 35) A = {d,a,b,c} B = {a,b,c,d} They both have 4 elements, thus they are equivalent. They have exactly the same elements, thus they are equal. Answer: equal and equivalent

37) A = {a,b} B = {c,b,a} They don’t have the same number of elements, thus they are not equivalent. They don’t have exactly the same elements, thus they are not equal. Answer: neither

39) A = {0,1,2,3,4} B = {a,b,c,d,e} They both have 5 elements, thus they are equivalent. They don’t have exactly the same elements, thus they are not equal. Answer: equivalent

41) A = {2,1} B = {1,4} They both have 2 elements, thus they are equivalent. They don’t have exactly the same elements, thus they are not equal. Answer: equivalent

43) A = {1} B = {2, 1} They don’t have the same number of elements, thus they are not equivalent. They don’t have exactly the same elements, thus they are not equal. Answer: neither

45) A = {1,3,5,7} B = {7,3,5,1} They both have 4 elements, thus they are equivalent. They have exactly the same elements, thus they are equal. Answer: equal and equivalent