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Step 1:
I will solve the first three problems from Section One.
Step 2:
Which of the following sets are equal?
Step 3:
: Find the solutions for the quadratic equations in set A and B.
So, $x = 2,1$.
Thus, set A = {1, 3}. Thus, set B = {1, 2}.
Step 4:
: Compare the sets found in Step 1.
Set A and set B are not equal since they do not have the same elements. Set A = {1, 3} and set B = {1, 2}.
Step 5:
: Compare the other sets.
Set C = {x | x ∈ ℕ, x < 3} = {1, 2} Set E = {1, 2} Set G = {3, 1} Set H = {1, 1, 3} Set C and set E are equal. Set G and set H are not equal to any other sets.
Final Answer
Set A ≠ Set B, Set A = {1, 3}, Set B = {1, 2}, Set C = Set E = {1, 2}, Set G ≠ {1, 2, 3}, Set H ≠ {1, 2, 3}. 2. List the elements of the following sets if the universal set is U = {a, b, c, ..., y, z}. Step 1: Find the elements of set A. Set A = {x |x is a vowel} So, set A = {a, e, i, o, u}. Step 2: Find the elements of set C. Set C = {x |x precedes f in the alphabet} So, set C = {a, b, c, d, e, g}. Step 3: Find the elements of set B. Set B = {x |x is a letter in the word “little”} So, set B = {l, i, t, t, l, e}. Step 4: Find the elements of set D. Set D = {x |x is a letter in the word “title”} So, set D = {t, i, t, l, e}. Step 5: Compare the sets. Set A ≠ Set C, Set A ≠ Set B, Set A ≠ Set D. Set A = {a, e, i, o, u}, Set C = {a, b, c, d, e, g}, Set B = {l, i, t, t, l, e}, Set D = {t, i, t, l, e}. 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? Step 1: Find the sets that can be equal to a set X under the given conditions. (a) X and B are disjoint. The only set that is disjoint from set B is set C. So, X = C. (c) X⊆A but X ⊈ C. Set D and set E can be equal to a set X under this condition. So, X = D or X = E. (b) X ⊆ D but X ⊈ B. Set E can be equal to a set X under this condition. So, X = E. (d) X⊆C but X ⊈ A. No set can be equal to a set X under this condition because all sets have elements in common with set A. X = C for condition (a), X = D or X = E for condition (c), X = E for condition (b), no set for condition (d).
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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 @ willyacadamia^2019@gmail.com.
Section One
1. Which of the following sets are equal?
A = {x | x^2 − 4x + 3 = 0}, C = {x | x ∈ N, x < 3}, E = {1, 2}, G = {3, 1}, B={x|x^2−3x+ 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.
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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 |x^2 = 9,2x = 8}.
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