AI Homework HelperFor Step-by-Step Solutions

LINEAR ALGEBRA II: Assignment No.2(Sets, Relations and Functions). AUGUST 2025: Date of Submission Wednesday 3rd September, 2025: mode of submission – hard copy or email @ willyacadamia2019@gmail.com. Section One 1. Which of the following sets are equal? A = {x | x2 − 4x + 3 = 0}, C = {x | x ∈ N, x < 3}, E = {1, 2}, G = {3, 1}, B={x|x2−3x+2=0}, D = {x|x∈N, x is odd, x<5}, F={1,2,1}, H={1,1,3}. Hint: for the quadratic Equations, get the values of x which shall be elements of set A and B.) 2. List the elements of the following sets if the universal set is U = {a, b, c, ..., y, z}. Furthermore, identify which of the sets, if any, are equal. A = {x |x is a vowel}, C = {x |x precedes f in the alphabet}, B = {x |x is a letter in the word “little”}, D = {x |x is a letter in the word “title”}. 3. Let A= {1,2,...,8,9}, B={2,4,6,8}, C={1,3,5,7,9}, D={3,4,5}, E={3,5}. Which of the these sets can equal a set X under each of the following conditions? (a) X and B are disjoint. (c) X⊆A but X ⊈ C. (b) X ⊆ D but X ⊈ B. (d) X⊆C but X ⊈ A. 4. Consider the universal set U = {1,2,3,...,8,9} and sets A={1,2,5,6}, B={2,5,7}, C={1,3,5,7,9}. Find: (a) A∩B and A∩C (b) A∪B and B∪C (d)A\BandA\C (f)(A∪C)\Band(B⊕C)\A (c)AC and CC (e) A⊕B and A⊕C 5. The formula A\B = A ∩ B C defines the difference operation in terms of the operations of intersection and complement. Find a formula that defines the union A ∪ B in terms of the operations of intersection and complement. 6. The Venn diagram in Fig. (a) shows sets A, B, C. Shade the following sets: (a) A\(B∪C); (b)AC∩(B∪C); (c)AC∩(C\B). ( Note you can draw different diagram for each answer to avoid shading overlapping and congestion.) 7. Write the dual of each equation: (a) A=(BC∩A)∪(A∩B) (b) (A∩B)∪(AC∩B)∪(A∩BC)∪(AC∩BC)=U 8. Use the laws in Table 1-1 to prove each set identity: (a) (A∩B)∪(A∩BC) = A (b) A∪B=(A∩BC)∪(AC∩B)∪(A∩B) Section Two 9. Determine which of the following sets are finite: (a) Lines parallel to the x axis. (c) Integers which are multiples of 5. (b) Letters in the English alphabet. (d) Animals living on the earth. 1 10. A survey on a sample of 25 new cars being sold at a local auto dealer was conducted to see which of three popular options, air-conditioning (A), radio (R), and power windows (W ), were already installed. The survey found: 15 had air-conditioning (A), 12 had radio (R), 11 had power windows (W), 5 had A and P , 9 had A and R, 3 had all three options. 4 had R and W, Find the number of cars that had:(a) only W; (b) only A; (c) only R; (d) R and W but not A; (e) A and R but not W; (f) only one of the options; (g) at least one option; (h) none of the options. 11. Find the power set P(A) of A={1,2,3,4,5}. 12. Given A = [{a,b},{c},{d,e,f}]. (a) List the elements of A. (b) Find n(A). (c) Find the power set of A. 13. Let S = {1, 2, ..., 8, 9}. Determine whether or not each of the following is a partition of S : (a) [{1,3,6},{2,8},{5,7,9}] (b) [{1,5,7},{2,4,8,9},{3,5,6}] (c) [{2,4,5,8},{1,9},{3,6,7}] (d) [{1,2,7},{3,5},{4,6,8,9},{3,5}] Section Three 14. Prove : 2+4+6+···+2n = n(n+1) Using Mathematical Induction 15. Let S={a,b,c},T={b,c,d}, and W={a,d}. Find S×T×W. 16. Find x and y where: (a)(x+2,4)= (5,2x+y); (b)(y−2, 2x+1)= (x−1, y+2). 17. Prove: A×(B∩C)=(A×B)∩(A×C) 18. Consider the relation R = {(1, 3), (1, 4), (3, 2), (3, 3), (3, 4)} on A = {1, 2, 3, 4}. (a) Find the matrix MR of R. (b) Find the domain and range of R. (c) Find R−1. (d) Draw the directed graph of R. 19. Determine if each function is one-to-one. (a) To each person on the earth assign the number which corresponds to his age. (b) To each country in the world assign the latitude and longitude of its capital. (c) To each book written by only one author assign the author. (d) To each country in the world which has a prime minister assign its prime minister. 20. Let functions f,g,h from V ={1,2,3,4} into V be defined by :f(n)= 6−n, g(n)=3, h = {(1, 2), (2, 3), (3, 4), (4, 1)}. Decide which functions are: (a) one-to-one; (b) onto; (c) both; (d) neither. 21. Prove Theorem 3.1: A function f : A → B is invertible if and only if f is both one-to-one and onto. 22. Find the cardinal number of each set: (a) {x | x is a letter in “BASEBALL”}; (b) Power set of A = {a,b,c,d,e}; (c) {x |x2 = 9,2x = 8}. 2

write the summary

what is the meaning of plagarism

LINEAR ALGEBRA II: Assignment No.2(Sets, Relations and Functions). AUGUST 2025: Date of Submission Wednesday 3rd September, 2025: mode of submission – hard copy or email @ willyacadamia2019@gmail.com. Section One 1. Which of the following sets are equal? A = {x | x2 − 4x + 3 = 0}, C = {x | x ∈ N, x < 3}, E = {1, 2}, G = {3, 1}, B={x|x2−3x+2=0}, D = {x|x∈N, x is odd, x<5}, F={1,2,1}, H={1,1,3}. Hint: for the quadratic Equations, get the values of x which shall be elements of set A and B.) 2. List the elements of the following sets if the universal set is U = {a, b, c, ..., y, z}. Furthermore, identify which of the sets, if any, are equal. A = {x |x is a vowel}, C = {x |x precedes f in the alphabet}, B = {x |x is a letter in the word “little”}, D = {x |x is a letter in the word “title”}. 3. Let A= {1,2,...,8,9}, B={2,4,6,8}, C={1,3,5,7,9}, D={3,4,5}, E={3,5}. Which of the these sets can equal a set X under each of the following conditions? (a) X and B are disjoint. (c) X⊆A but X ⊈ C. (b) X ⊆ D but X ⊈ B. (d) X⊆C but X ⊈ A. 4. Consider the universal set U = {1,2,3,...,8,9} and sets A={1,2,5,6}, B={2,5,7}, C={1,3,5,7,9}. Find: (a) A∩B and A∩C (b) A∪B and B∪C (d)A\BandA\C (f)(A∪C)\Band(B⊕C)\A (c)AC and CC (e) A⊕B and A⊕C 5. The formula A\B = A ∩ B C defines the difference operation in terms of the operations of intersection and complement. Find a formula that defines the union A ∪ B in terms of the operations of intersection and complement. 6. The Venn diagram in Fig. (a) shows sets A, B, C. Shade the following sets: (a) A\(B∪C); (b)AC∩(B∪C); (c)AC∩(C\B). ( Note you can draw different diagram for each answer to avoid shading overlapping and congestion.) 7. Write the dual of each equation: (a) A=(BC∩A)∪(A∩B) (b) (A∩B)∪(AC∩B)∪(A∩BC)∪(AC∩BC)=U 8. Use the laws in Table 1-1 to prove each set identity: (a) (A∩B)∪(A∩BC) = A (b) A∪B=(A∩BC)∪(AC∩B)∪(A∩B) Section Two 9. Determine which of the following sets are finite: (a) Lines parallel to the x axis. (c) Integers which are multiples of 5. (b) Letters in the English alphabet. (d) Animals living on the earth. 1 10. A survey on a sample of 25 new cars being sold at a local auto dealer was conducted to see which of three popular options, air-conditioning (A), radio (R), and power windows (W ), were already installed. The survey found: 15 had air-conditioning (A), 12 had radio (R), 11 had power windows (W), 5 had A and P , 9 had A and R, 3 had all three options. 4 had R and W, Find the number of cars that had:(a) only W; (b) only A; (c) only R; (d) R and W but not A; (e) A and R but not W; (f) only one of the options; (g) at least one option; (h) none of the options. 11. Find the power set P(A) of A={1,2,3,4,5}. 12. Given A = [{a,b},{c},{d,e,f}]. (a) List the elements of A. (b) Find n(A). (c) Find the power set of A. 13. Let S = {1, 2, ..., 8, 9}. Determine whether or not each of the following is a partition of S : (a) [{1,3,6},{2,8},{5,7,9}] (b) [{1,5,7},{2,4,8,9},{3,5,6}] (c) [{2,4,5,8},{1,9},{3,6,7}] (d) [{1,2,7},{3,5},{4,6,8,9},{3,5}] Section Three 14. Prove : 2+4+6+···+2n = n(n+1) Using Mathematical Induction 15. Let S={a,b,c},T={b,c,d}, and W={a,d}. Find S×T×W. 16. Find x and y where: (a)(x+2,4)= (5,2x+y); (b)(y−2, 2x+1)= (x−1, y+2). 17. Prove: A×(B∩C)=(A×B)∩(A×C) 18. Consider the relation R = {(1, 3), (1, 4), (3, 2), (3, 3), (3, 4)} on A = {1, 2, 3, 4}. (a) Find the matrix MR of R. (b) Find the domain and range of R. (c) Find R−1. (d) Draw the directed graph of R. 19. Determine if each function is one-to-one. (a) To each person on the earth assign the number which corresponds to his age. (b) To each country in the world assign the latitude and longitude of its capital. (c) To each book written by only one author assign the author. (d) To each country in the world which has a prime minister assign its prime minister. 20. Let functions f,g,h from V ={1,2,3,4} into V be defined by :f(n)= 6−n, g(n)=3, h = {(1, 2), (2, 3), (3, 4), (4, 1)}. Decide which functions are: (a) one-to-one; (b) onto; (c) both; (d) neither. 21. Prove Theorem 3.1: A function f : A → B is invertible if and only if f is both one-to-one and onto. 22. Find the cardinal number of each set: (a) {x | x is a letter in “BASEBALL”}; (b) Power set of A = {a,b,c,d,e}; (c) {x |x2 = 9,2x = 8}. 2

write the summary

what is the meaning of plagarism