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Step 1:
So, $$x = 1, 2
To check which sets are equal, we first find the solutions for the quadratic equations in sets A and B. Now, let's compare the sets: A = {x | x^2 − 4x + 3 = 0} = {1, 3} B = {x | x^2 − 3x + 2 = 0} = {1, 2} C = {x | x ∈ N, x < 3} = {1, 2} E = {1, 2} G = {3, 1} We can see that sets A, C, E, and G are equal, and set B is not equal to any of the other sets.
Final Answer
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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.
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}]
2. To list the elements of the given sets: A = {x |x is a vowel} = {a, e, i, o, u} C = {x |x precedes f in the alphabet} = {a, b, c, d, e, g} B = {x |x is a letter in the word “little”} = {l, i, t, t, l, e} = {l, i, t, e} D = {x |x is a letter in the word “title”} = {t, i, t, l, e} Since sets B and D have the same elements, they are equal. 3. To find the sets that can be equal to a set X under the given conditions: (a) X and B are disjoint: No set can be equal to X since all sets have elements in common with set B. (c) X ⊆ A but X ⊈ C: Set D can be equal to X. (b) X ⊆ D but X ⊈ B: No set can be equal to X since all sets have elements in common with set B. (d) X ⊆ C but X ⊈ A: Set E can be equal to X. 4. To find the required sets: (a) A ∩ B = {2, 5}, A ∩ C = {1, 5} (b) A ∪ B = {1, 2, 5, 6, 7}, B ∪ C = {1, 2, 3, 4, 5, 7, 9} (d) A \ B = {1, 6}, A \ C = {1, 2, 6, 9} (f) (A ∪ C) \ B = {1, 3, 6, 7, 9}, (B ⊕ C) \ A = {3, 4, 6, 8, 9} (c) A C = {5}, C C = {5, 7} (e) A ⊕ B = {1, 2, 3, 4, 6, 7}, A ⊕ C = {1, 2, 3, 4, 6, 7, 8, 9} 5. A ∪ B = (A ∩ B) ∩ (A ∩ B)' + A ∩ B 6. Shading the sets: (a) A\(B∪C): Shade everything in A except the parts that overlap with B or C. (b) AC ∩ (B ∪ C): Shade the intersection of sets A and C, and then shade the parts of B or C that overlap with the shaded area. (c) AC ∩ (C\B): Shade the intersection of sets A and C, and then shade the parts of C that do not overlap with B. 7. Dual of each equation: (a) A = (B'C)' ∩ (A ∩ B')' (b) U = (A ∩ B)' ∩ (A ∩ B)' ∩ (A ∩ B)' 8. Using the laws in Table 1 - 1 to prove each set identity: (a) (A ∩ B) ∪ (A ∩ BC) = A LHS: (A ∩ B) ∪ (A ∩ BC) = A ∩ (B ∪ BC) (Distributive law) = A ∩ U (Complement law) = A (Identity law) (b) A ∪ B = (A ∩ BC) ∪ (AC ∩ B) ∪ (A ∩ B) LHS: A ∪ B = (A ∪ B) ∩ U (Identity law) = (A ∪ B) ∩ ((B ∩ C) ∪ (B ∩ C)') (Complement law) = (A ∩ (B ∩ C)') ∪ (B ∩ (B ∩ C)') ∪ (A ∩ B) (Distributive law) = (A ∩ BC)' ∪ (B ∩ BC)' ∪ (A ∩ B) (Complement law) = (A ∩ BC)' ∪ (AC ∩ B)' ∪ (A ∩ B) (Distributive law) The proofs for (a) and (b) are complete. I'll continue solving the remaining questions in the next part.
