Sets Worksheet Name:MA 201WARNING: You must SHOW ALL OF YOUR WORK. You will receive NO CREDIT if youdo not show your work.1. Let U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} be the universe. Let A = {x ∈ U| x is even},B = {x ∈ U| 1 ≤ x < 10}, and C = {x ∈ U| x is even or x = 15}. Find the following.(a) n(A)(b) n(B)(c) n(A ∪ B)(d) n(A ∩ B)(What relationship do you notice between n(A), n(B), n(A ∪ B), and n(A ∩ B)? Doesthis relationship always hold? If so, explain why it always holds. If not, provide acounterexample.)(e) A ∩ B(f) A ∪C(g)A ∪ C(h) (A ∪ B) ∩ C2. True or False.(a) {1, 2} = {2, 1}(b) {1, 2} ∼ {2, 1}(c) {1, 2} ∼ {3, 4}(d) (1, 2) = (2, 1)(e) Let B = {a, b, c, d, e}. Then B = 5.(f) ∅ ⊆ {a, b, c}(g) 0 ÷ 21 is defined.(h) 21 ÷ 0 is defined.(i) {a, b, c} ∪ ∅ = {a, b, c}(j) {a, b, c} ∩ ∅ = {a, b, c}(k) ∅ = {0}(l) n(∅) = 0(m) Let A = {x| x is an even whole number}.Let B = {y| y is an even natural number}. Then B ⊆ A.(n)Let A = {x| x is an even whole number}.Let B = {y| y is an even natural number}. Then B ⊂ A.13. True or False. If the statement is true, briefly explain why it is true. If it is false, provide acounterexample.(a) If A and B are finite sets, then n(A) + n(B) = n(A ∪ B).(b) If A and B are finite sets, then n(A) × n(B) = n(A × B)(c) If n(A ∩ B) < n(A), then B ⊂ A.(d) If A ⊆ B and B ⊂ C, then A ⊂ C.4. (a) Show that the set of whole numbers, W , is equivalent to the set of natural numbers, N,by carefully describing a one-to-one correspondence between the sets.(b) According to the one-to-one correspondence you described in part (a), which whole num-ber is paired with the natural number 999?(c) According to the one-to-one correspondence you described in part (a), which naturalnumber is paired with the whole number 999?(d) According to the one-to-one correspondence you described in part (a), which whole num-ber is paired with the natural number x?(e) According to the one-to-one correspondence you described in part (a), which naturalnumber is paired with the whole number y?5. (a) Show that the set of even whole numbers, E, is equivalent to the set of odd wholenumbers, O, by carefully describing a one-to-one correspondence between the sets.(b) According to the one-to-one correspondence you described in part (a), which even numberis paired with the odd number 999?(c) According to the one-to-one correspondence you described in part (a), which odd numberis paired with the even number 764?(d) According to the one-to-one correspondence you described in part (a), which even numberis paired with the odd number m?(e) According to the one-to-one correspondence you described in part (a), which odd numberis paired with the even number n?6. Do number 11 on page 94 of your textbook.7. Let A = {a, b, c} and B = {c, d, ef}. Then n(A) = 3, n(B) = 4, and n(A ∪ B) = 6 (Why?)Look at the definition for addition of whole numbers given on page 99 of your textbook. Inthis example, n(A) + n(B) 6= n(A ∪ B). Is there a conflict between this example and thedefinition for addition of whole numbers? Why or why not? Explain briefly.8. Use the measurement model to illustrate the following.(a) 4 + 6 = 6 + 4(b) 4 × (2 + 3) = 4 × 2 + 4 × 39. True or False. Briefly justify your answer.2(a) {0, 1} is closed under multiplication.(b) {0, 1} is closed under subtraction.(c) {0, 1} is closed under addition.(d) Let X be s subset of the whole numbers that contains 2 and 4. If X isclosed under addition then 3 cannot be an element of X.10. Let X be a subset of the whole numbers that contains 2. If X is closed under addition, whatwhole numbers must be contained in X? What, if any, numbers are certainly not containedin X?11. Use the number line to illustrate the following facts.(a) 15 − 2 = 13(b) 4 ∗ 5 = 2012. Do number 16 on page 109 of your textbook.13. Do number 28 on page 111 of your textbook.14. State the Division Algorithm.15. Use sets to show that 6 < 9.16. Do units need to be the same when adding? when subtracting? when multiplying? whendividing?17. For each subtraction model, write a separate word problem that illustrates53 − 26.(a) The comparison model for subtraction(b) The missing addend model for subtraction(c) The take-away model for subtraction(d) The measurement model for subtraction18. For each addition model, write a separate word problem that illustrates53 + 26.(a) The set model for addition(b) The measurement model for addition19. For each multiplication model, write a separate word problem that illustrates4 × 3.(a) The repeated addition model for multiplication3(b) The array model for multiplication(c) The rectangular area model for multiplication(d) The multiplication tree model for multiplication(e) The Cartesian product model for multiplication20. For each division model, write a separate word problem that illustrates35 ÷ 7.(a) The repeated subtraction model for division(b) The partition model for division21. Clearly explain why division by zero is undefined.22. Clearly explain why0253is defined.23. Rewrite as a single whole number exponential, if possible.(a)71574(b) 25− 23(c) (25)6(d) 45× 6524. Evaluate 30.25. Do number 25 on page 131 of your textbook.26. Let X = {1, 2, 3} and Y = {a, b}.(a) Find X × Y .(b) Find Y × X.(c) Is X × Y = Y × X?(d) Is n(X × Y ) = n(Y × X)?27. Let X = {a, b} and Y = {b, a}.(a) Find X × Y .(b) Find Y × X.(c) Is X × Y = Y × X?(d) Is n(X × Y ) = n(Y × X)?28. Let A and B be finite sets.(a) What can be said about n(A × B) and n(B × A)? Relate this observation to a propertyof whole number multiplication.(b) If A × B = B × A, what must be true about A and