BEE3054 Problem Set 3
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Problem Set 3
Sets
Module number: BEE3054
Problem 1. Let A = {1, 2, 3, 4}, B = {3, 4, 5, 6, 7}, and C = {2, 3, 8, 9}. Please write down:
A U B =
A U C =
B U C =
A n B =
A n C =
B n C =
A U (B n C) =
(A U B) n C =
Problem 2. Prove the following theorem:
Theorem 1. For any sets A and B , A n B C A C A U B and A n B C B C A U B
Problem 3. Prove the following theorem:
Theorem 2. For any sets A, B, and C, A n (B U C) = (A n B) U (A n C) .
Use a Venn diagram and the sets from Problem 1 to demonstrate the result.
Problem 4. Prove the following theorem:
Theorem 3. For any sets A and B , (A U B) / (A n B) = (A / B) U (B / A) .
Problem 5. A consumer survey of magazine purchases found that of the respondents,
. 25 purchased Newsweek
. 26 purchased Time
. 26 purchased Fortune
. 9 purchased both Newseek and Fortune
. 11 purchased both Newsweek and Time
● 8 purchased both Time and Fortune
● and 3 purchased all three.
1. Find the number of people who purchase at least one of the three magazines by intro- ducing and using mathematical notation to provide the calculation.
2. Represent all consumers and their purchases with a Venn diagram.
3. Find the number of people who purchase exactly one magazine.
Problem 6. Let A = {{a, b}, {c}, {d, e, f}}.
1. List the elements of A.
2. Are the following true or false?
(a) a e A.
(b) {a} e A.
(c) c e A.
(d) {c} e A.
(e) {d, e, f} S A.
(f) {{a, b}} S A.
(g) 0 e A
(h) 0 S A
3. Write out p(A).
Problem 7. Let U be an arbitrary set and Xi for i = 1, . . . , n be subsets satisfying (a) Xi n Xj = 0 for i e {1, . . . , n}, j e {1, . . . , n}, i 
j and (b) X1 u X2 u . . . u Xn = U . Construct a set s S p(U) which satisfies both properties and explain using a graph and words.
Problem 8. Prove that for any sets A, B , C, and D ,
(A × B) n (C × D) = (A n C) × (B n D)
where × is the cross product (also known as the “Cartesian product”). Support your proof with a Venn diagram.
Problem 9. Let P denote the price and Q the quantity produced of some commodity. The demand set is defined by D = {(P, Q) e R+ × R+ |Q = 4 - P2 } where (P, Q) is an ordered pair. The supply set is defined as S = {(P, Q) e R+ × R+ |Q = 4P - 1}. Show graphically and construct the sets
D u S
D n S
and provide an economic interpretation of these sets.
Problem 10. Let f be defined by
f :Z → Z
之 → |之 |
Provide a proof of whether the inverse f − 1 exists and provide the inverse if it does.
Problem 11. Let Ⅹ 三 (1．2．3．4) . Check whether the following relations are functions from Ⅹ into Ⅹ .
1. f 三 ((2．3)．(1．4)．(2．1)．(3．2)．(4．4))
2. g 三 ((3．1)．(4．2)．(1．1))
3. h 三 ((2．1)．(3．4)．(1．4)．(2．1)．(4．4))
Problem 12. Let f : R → R be defined by f (z) = 2z + 1 and g : R → R be defined by g(z) = z2 - 2. Let the notation f 。g : R → R be the function defined by f (g(z)). Show that f 。g 
g 。f .
Problem 13. Let Ⅹ 三 [-1．1] be the closed interval from -1 to 1. Consider the following functions from Ⅹ → Ⅹ .
f (z) 三 sin(z)
g(z) 三 sin(πz)
h(z) 三 sin _ 
z、
For each function, check whether it is (i) injective, (ii) surjective, and (iii) bijective.
Problem 14. Let f : Ⅹ → Y be a bijective function. Provide a function h : p(Ⅹ) → p(Y) that is also bijective using f , where p is the power set.
2023-02-20