Mat142 Problems and Answers

Section 1.1 Section 1.1 answers: 1) A = {2,4,6,8,10} 3) B = {2,4,6,8} 5) A = {4} 7) B = {chocolate, vanilla, strawberry} 9) no 11) yes 13) yes 15) no 17) finite 19) infinite 21) finite 23) infinite 25) n(B) = 6 27) n(A) = 3 29) n(A) = 8 31) n(A) = 3 33) n(C) = 5 35) Equal and equivalent 37) neither 39) equivalent 41) equivalent 43) neither 45) equal and equivalent 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. 2) The set A of odd numbers between 1 and 10 inclusive. 3) The set B of even numbers between 1 and 10 exclusive. 4) The set B of odd numbers between 1 and 10 exclusive. 5) The set A of solutions to the problem: x + 6 = 10 6) The set A of solutions to the problem x – 4 = 9 7) The set B of flavors of ice cream in Neapolitan ice cream. 8) The set B of days of the week. 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. 10) B is the set of GCC students whose GPA is 3.0 or greater. 11) D is the set of numbers whose square is 16. 12) C is the set of good GCC students. 13) The set of years that the Arizona Cardinals won their division. 14) The set of Great Lakes. 15) The set of cities that have nice climates. 16) The set of good smart phones Homework #17 – 24. Determine whether each set is finite or infinite. 17) 18) 19) 20) The set of factors of 12. The set of factors of 24. The set of multiples of 3. The set of multiples of 5. 21) 22) 23) 24) The set of positive multiples of 4 that are less than 30. The set of multiples of 4 that are greater than 10 and less than 50. The set of even positive integers. The set of odd positive integers. Homework #25 – 34. Find the cardinal number of the following sets. That is find the number of elements in each set. 25) 26) 27) 28) 29) 30) 31) 32) 33) 34) The set B of natural numbers between 5 and 10 inclusive. The set B of even natural numbers between 5 and 10 inclusive. The set A of even natural numbers greater than 4 and less than 11. The set A of odd natural numbers greater than 2 and less than or equal to 5. The set A of natural numbers less than 9. The set A of natural numbers less than or equal to 5. Find n(A) where A = {1,3,5} Find n(B) where B = {a, c, e, o, v} Find n(C) where C = {2,4,6,8,10} Find n(D) where D = {0} 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} 36) A = {1,2,3,4} B = {4,3,2} 37) A = {a,b} B = {c,b,a} 38) A = {a,b} B = {b, a} 39) A = {0,1,2,3,4} B = {a,b,c,d,e} 40) A = {1,3,2} B = {lemon, cherry, apple} 41) A = {2,1} B = {1,4} 42) A = {1,2,3} B = {4,5,6,7} 43) A = {1} B = {2, 1} 44) A = {1,2,3,4,5} B = {4,5,3,2,1} 45) A = {1,3,5,7} B = {7,3,5,1} Section 1.2 Section 1.2 answers: 1) D equals the set of natural numbers greater than or equal to two. 3) A equals the set of natural numbers between 1 and 7 inclusive 5) D = the set of the seven dwarfs 7) V = the set of vowels 9) D = {𝑥|𝑥 ∈ 𝑁, 𝑥 ≥ 2} 11) A = { x | x is a Natural number and 1 ≤ 𝑥 ≤ 7 } 13) P = { x | x is a Natural number and 3 ≤ 𝑥 ≤ 11 } 15) P = { x | x is a Natural number and 3 < 𝑥 < 11 } 17) A = {5,6,7,8,9,10} 19) A = {6,8,10…} 21) A = {5,6,7…} 23) A = { } 25) B = {5} 27) C = {1,2} 29) D = {3,4} 31) D = {3,4,5} Homework #1 – 8. Write a word description of each set 1) D = {2, 3, 4, …} 2) B = {4,5,6 …} 3) A = {1,2,3,4,5,6,7} 4) E = {5,6,7,8,9,10} 5) D = {Bashful, Doc, Dopey, Grumpy, Happy, Sleepy, Sneezy} 6) B = {California, Oregon, Washington} 7) V = {a, e, i, o, u} 8) T = {1,4,9,16,25,36,…} Homework #9 – 16: Write the sets in set-builder form. (There are many correct answers.) 9) D = {2, 3, 4, …} 10) A = {3,4,5,6…} 11) A = {1,2,3,4,5,6,7} 12) B = {1,2,3,4,5,6} 13) P = set of natural numbers between 3 and 11 inclusive 14) P = set of natural numbers between 4 and 9 inclusive 15) P = set of natural numbers between 3 and 11 exclusive 16) P = set of natural numbers between 4 and 9 exclusive Homework # 17 – 32. Write sets in roster form. 17) The set A of natural numbers between 5 and 10 inclusive. 18) The set A of even natural numbers between 5 and 10 inclusive. 19) The set A of even natural numbers greater than 4. 20) The set A of odd natural numbers greater than or equal to 5. 21) The set A of natural numbers greater than 4. 22) The set A of natural numbers greater than 5. 23) The set A of natural numbers that are negative. 24) The A set of natural numbers less than 0. 25) B = { k | k+1=6 } 26) B = { x | x – 3 = 5 } 27) C = { m | m is a natural number, m < 3} 28) C = { m | m is a natural number, m ≤ 3} 29) D = {m | m ∈ 𝑁, 2 < m < 5} 30) D = { m | m ∈ 𝑁, 2 ≤ m < 5} 31) D = { m | m ∈ 𝑁, 2 < m ≤ 5} 32) D = { m | m ∈ 𝑁, 3 < m < 7} Section 1.3: Subsets Section 1.3 answers: 1) false, silver is not a set 3) true 5) true 7) false “a” is not a set 9) true 11) true 13) false, they are equal and the proper subset symbol does not allow sets to be equal. 15) false, 5 is in the left set but not in the right set 17) true 19) B ⊆ A, B⊂A 21) A = B, A ⊆ B, B⊆A 23) A ⊆ B, A ⊂ B 25) none 27) B ⊆ A, B⊂A 29) B ⊆ A, B ⊂ A 31) A = B, A ⊆ B, B ⊆ A 33) A ⊆ B, A ⊂ B Homework #1-18: Answer yes or no, if no give the reason 1) Is A ⊆ B given A = silver, B = {gold, silver, diamond} 2) Is A⊆ B given A = Phoenix, B = {Phoenix, Glendale, Peoria, Scottsdale} 3) Is C ⊆ D given C = {Phoenix}, D = {Phoenix, Glendale, Peoria, Scottsdale} 4) Is C ⊆ D given C = {silver}, D = { gold, silver, diamond} 5) Is A ⊆ B given A = {2,3} , B = {1,2,3,4,5} 6) Is A ⊆ B given A = 2,3 B = {1,2,3,4,5} 7) Is A ⊆ B given A = a, B = {𝑥|𝑥 𝑖𝑠 𝑎 𝑣𝑜𝑤𝑒𝑙} 8) Is A⊆ B given A= {a}, B = {𝑥|𝑥 𝑖𝑠 𝑎 𝑣𝑜𝑤𝑒𝑙} 9) Is A ⊆ B given A = { }, B = {1,2,3,4,5} 10) Is S ⊆ T given S = ∅, T ={1,2,3} 11) Is S ⊂ T given S= ∅, T = {1,2,3,4,5} 12) Is S ⊂ T given S = { }, T = {1,2} 13) Is A ⊂ B given A = {1,2,3}, B = {3,2,1} 14) Is A ⊂ B given A = {a,b,c,d}, B = {d,c,b,a} 15) Is C ⊂ D given C = {1,2,3,4,5}, D = {1,2,3,4} 16) Is C ⊂ D given C = {w,x,y}, D = {x,y,z} 17) Is A ⊂ B given A = { 4,3,2,1 }, B = {1,2,3,4,5} 18) Is A ⊂ B given A = { e,b,c,d,a }, B = {a,b,c,d,e,f} Homework #19 – 34: Determine which of these are true. (Choose every answer that is true, in many instances there will be more than one correct choice.) A = B, A ⊆ B, B ⊆ A, A ⊂ B, B ⊂ A, or none of these 19) A = {Trix, Captain Crunch, Rice Krispees} 20) A = {red, white, blue} 21) A = {5,7,9} B = {Rice Krispees} B = {red, blue} B = {9, 5, 7} 22) A = {a,b,c} B = {c, b, a} 23) A = {2,4,6} 24) A = {2,5,7,9} B = {7,9} 25) A = {a,b,c} B = {a,b,d} 26) A = {c,e,g,h} B = {c,b,a} 27) A = {𝑥|𝑥 ∈ 𝑁 𝑎𝑛𝑑 𝑥 < 9} B = {4,5,6} 28) A = {𝑥|𝑥 ∈ 𝑁 𝑎𝑛𝑑 𝑥 < 5} B = {1,2,3} 29) A = {8,9,10,11…} B = {𝑥|𝑥 ∈ 𝑁 𝑎𝑛𝑑 𝑥 ≥ 9} 30) A = {𝑥|𝑥 ∈ 𝑁 𝑎𝑛𝑑 𝑥 ≥ 5} B = {4,5,6,7…} 31) A = {𝑥|𝑥 ∈ 𝑁 𝑎𝑛𝑑 2 < 𝑥 < 9} B = {3,4,5,6,7,8} 32) A = {𝑥|𝑥 ∈ 𝑁 𝑎𝑛𝑑 0 < 𝑥 < 5} B = {1,2,3,4} 33) A = {𝑥|𝑥 ∈ 𝑁 𝑎𝑛𝑑 2 < 𝑥 < 9} B = {0,1,2,3,4,5,6,7,8} 34) A = {𝑥|𝑥 ∈ 𝑁 𝑎𝑛𝑑 0 < 𝑥 < 5} B = {0,1,2,3,4} B = {2,4,6,8} Section 1.4 Section 1.4 answers: 1) {a,b} 3) {a,c,d,e} 5) {c,d} 7) {a} 9) {a,b,c,d} 11) {4,5} 13) {1,2,3,5} 15) ∅ 17) {5} 19) {4,5} 21) {1} 23) {1,2,3,5} 25) {1,2,3,5} 27) {1,2,3} 29) {4} 31) {2,3,4,6} 33) {b,c} 35) {b,c,d} 37) {b,c,d} 39) {a,b,c} 41) ∅ 43) ∅ 45) {2,4,6} 47) {3,5} 49) {6} 51) {1,2} 53) {1,2,3,4,5} 55) {1,2,3,4,5,6} #1-10: Find the following sets. U = {a,b,c,d,e} A = {c,d,e} B = {a,c,d} 1) A’ 2) B’ 3) 𝐴 ∪ 𝐵 4) 𝐴′ ∪ 𝐵′ 5) 𝐴 ∩ 𝐵 6) 𝐴′ ∩ 𝐵′ 7) 𝐴′ ∩ 𝐵 8) 𝐴 ∩ 𝐵′ 9) 𝐴′ ∪ 𝐵 10) 𝐴 ∪ 𝐵′ #11-20: Find the following sets. U = {1,2,3,4,5} A = {1,2,3} B = {5} 11) A’ 12) B’ 13) 𝐴 ∪ 𝐵 14) 𝐴′ ∪ 𝐵′ 15) 𝐴 ∩ 𝐵 16) 𝐴′ ∩ 𝐵′ 17) 𝐴′ ∩ 𝐵 18) 𝐴 ∩ 𝐵′ 19) 𝐴′ ∪ 𝐵 20) 𝐴 ∪ 𝐵′ #21-32: Find the following sets. U = {1,2,3,4,5,6} A = {1,2,3} B = {2,3,4} C = {1,5} 21) 𝐴 ∩ 𝐶 22) 𝐵 ∩ 𝐶 23) 𝐴 ∪ 𝐶 24) 𝐵 ∪ 𝐶 25) 𝐴 ∩ 𝐵 ∪ 𝐶 26) 𝐴 ∪ 𝐵 ∩ 𝐶 27) 𝐵 ∪ 𝐶 ∩ 𝐴 28) 𝐵 ∩ 𝐴 ∪ 𝐶 29) 𝐴′ ∩ 𝐵 30) 𝐴 ∩ 𝐵′ 31) 𝐴′ ∪ 𝐵 ∩ 𝐶′ 32) 𝐵′ ∩ 𝐴 ∪ 𝐶′ #33-44: Find the following sets. U = {a,b,c,d} A = {a,b,c} B = {b,c,d} C = {a,d} 33) 𝐴 ∩ 𝐶′ 34) 𝐵′ ∩ 𝐶 35) 𝐴′ ∪ 𝐶′ 36) 𝐵′ ∪ 𝐶′ 37) 𝐴′ ∩ 𝐵 ∪ 𝐶′ 38) 𝐴′ ∪ 𝐵′ ∩ 𝐶 39) 𝐵′ ∪ 𝐶′ ∩ 𝐴 40) 𝐵′ ∩ 𝐴′ ∪ 𝐶 41) 𝐴′ ∩ 𝐵′ 42) 𝐴 ∩ 𝐵′ 43) 𝐴′ ∪ 𝐵′ ∩ 𝐶′ 44) 𝐵 ∩ 𝐴′ ∪ 𝐶′ #45 – 56: Find the following sets. U = {1,2,3,4,5,6} S = {2,4,6} T = {1,2,4} V = {4,5,6} 45) 𝑆 ∪ (𝑇 ∩ 𝑉) 46) (𝑆 ∪ 𝑇) ∩ 𝑉 47) (𝑆 ∪ 𝑇)′ 48) (𝑉 ∪ 𝑆)′ 49) 𝑆 ∩ (𝑉 ∩ 𝑇 ′ ) 50) (𝑆 ′ ∩ 𝑉 ′ ) ∪ 𝑇 51) (𝑆 ′ ∪ 𝑉 ′ ) ∩ 𝑇 52) 𝑆′ ∪ 𝑇 ∩ 𝑉′ 53) 𝑇 ∪ 𝑉′ ∪ 𝑆′ 54) 𝑇 ∪ 𝑉′ ∩ 𝑆′ 55) (𝑉 ∩ 𝑇)′ ∪ 𝑆 56) 𝑉 ∪ (𝑆 ∩ 𝑇)′ Section 1.5: Venn Diagrams – Shading Section 1.5 answers: 1) 3) 5) 7) 9) 11) 13) 15) 17) 19) 21) 23) Homework #1-9: Sketch the region. 1) 𝐴 ∩ 𝐵′ 2) 𝐴′ ∩ 𝐵 3) 𝐴′ ∪ 𝐵 4) 𝐴 ∪ 𝐵′ 5) (𝐴 ∩ 𝐵)′ 6) (𝐴 ∩ 𝐵 ′ )′ 7) (𝐴′ ∪ 𝐵)′ 8) (𝐴 ∪ 𝐵 ′ )′ 9) 𝐴′ ∪ 𝐵′ Homework #10-23: Sketch the shaded region. 10) 𝐴 ∩ 𝐵′ ∪ 𝐶 11) 𝐴 ∩ 𝐵 ∪ 𝐶′ 12) 𝐴 ∩ 𝐵′ ∩ 𝐶 13) 𝐴 ∩ 𝐵 ∩ 𝐶′ 14) 𝐴 ∪ 𝐵 ∪ 𝐶′ 15) 𝐴′ ∪ 𝐵 ∪ 𝐶 16) (𝐵 ∩ 𝐶)′ ∪ 𝐴 17) (𝐴 ∩ 𝐵)′ ∪ 𝐶 18) 𝐵 ∪ (𝐴 ∩ 𝐶) 19) 𝐴 ∪ (𝐵 ∩ 𝐶) 20) 𝐵 ∩ (𝐴 ∪ 𝐶) 21) 𝐴 ∩ (𝐵 ∪ 𝐶) 22) 𝐵 ∩ (𝐴 ∪ 𝐶)′ 23) 𝐴 ∩ (𝐵 ∪ 𝐶)′ Section 1.6: Section 1.6 answers: 1) S = {1,4,9,16} E = {2,4,6,8,10,12,14,16,18,20} U = {1,2,3,4,…18,19,20} 3) A = {a,b,c} B = {c,d,e,f} ξ = {a,b,c,d,e,f,g,h,i} 5) A = {4,6,8,10,12,14,16,18} B = { 6,9,12,15,18} C = 5,6,10,15} ξ = {4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19} 7) A = {1,2,4,5} 9) B = {4,5,7,8,10} C = {2,3,4,7,8,9} U = {1,2,3,4,5,6,7,8,9,10} 9) {d,e,f,g,h,i} 11) {c,d,e,f,g,h,I,j} 13) {a,b,d,e,f,g,h,I,j} 15) {a,b} 17) {a,b,d,e,f,g,h,I,f} 19) {8} 21) {5,6,7,9} 23) {4,5,6,7,8,9} 25) {5,6,7,9} 27) {1,3,5,6,7,10} 29) {4} 31) {1,2,4,5,6,7,8,9,10} 33) {2,3,4,5,8,9,10,15,20} 35) {2,4,6,8,12,15,20} 37) {6,12,20} 39) {2,4,8} Homework #1-8: 1) Here is a Venn diagram that represents sets U, S and E. Write down the elements of each set U={ } S={ } E={ } 2) Here is a Venn diagram that represents the sets A, B and E (the universal set is called E in this problem, this is unusual, but not wrong). Write down the elements of each set. E={ } A={ } B={ } 3) Here is a Venn diagram that represents sets ξ, A and B (the universal set is called ξ (pronounced xee, instead of U). Write down the elements of each set U={ } A={ } B={ } 4) Write down the elements of each set depicted in the Venn diagram. ξ ={ } A={ } B={ } 5) Write down the elements of each set depicted in the Venn diagram. ξ ={ } B={ } C={ } A={ } 6) Write down the elements of each set depicted in the Venn diagram. ξ ={ } B={ } C={ } A={ } 7) Write down the elements of each set depicted in the Venn diagram. U={ } B={ } C={ } A={ } 8) Write down the elements of each set depicted in the Venn diagram. U={ } B={ } C={ } A={ } Homework # 9 – 17. Use the Venn diagram to find the requested sets. 9) A’ 10) B’ 11) 𝐴′ ∪ 𝐵 12) 𝐴 ∪ 𝐵′ 13) (𝐴 ∩ 𝐵)′ 14) (𝐴 ∩ 𝐵 ′ )′ 15) (𝐴′ ∪ 𝐵)′ 16) (𝐴 ∪ 𝐵 ′ )′ 17) 𝐴′ ∪ 𝐵′ Homework # 18 – 25. Use the Venn diagram to find the requested sets. 18) 𝐴 ∩ 𝐵′ 19) 𝐴′ ∩ 𝐵′ 20) 𝐴′ ∩ 𝐵 21) 𝐴′ ∩ 𝐵 22) (𝐴 ∩ 𝐵)′ 23) (𝐴 ∩ 𝐵 ′ )′ 24) (𝐴′ ∪ 𝐵)′ 25) (𝐴 ∪ 𝐵 ′ )′ Homework # 26 – 31. Use the Venn diagram to find the requested sets. 26) 𝐴 ∩ 𝐵 ∪ 𝐶′ 27) 𝐴 ∩ 𝐵′ ∪ 𝐶 28) 𝐴 ∩ 𝐵′ ∩ 𝐶 29) 𝐴 ∩ 𝐵 ∩ 𝐶′ 30) 𝐴 ∪ 𝐵 ∪ 𝐶′ 31) 𝐴′ ∪ 𝐵 ∪ 𝐶 Homework # 32 – 39. Use the Venn diagram to find the requested sets. 32) (𝐴 ∩ 𝐶)′ ∪ 𝐵 33) (𝐴 ∩ 𝐵)′ ∪ 𝐶 34) 𝐵 ∪ (𝐴 ∩ 𝐶) 35) 𝐴 ∪ (𝐵 ∩ 𝐶) 36) 𝐵 ∩ (𝐴 ∪ 𝐶) 38) 𝐵 ∩ (𝐴 ∪ 𝐶)′ 39) 𝐴 ∩ (𝐵 ∪ 𝐶)′ 37) 𝐴 ∩ (𝐵 ∪ 𝐶)