(a) For A = {1, 2, 3}, give an example of three subsets A1,A2 and A3 of A such that

Write each of the following sets by listing its elements.(a) A = {x Q : x4 4 = 0}.(b) B = nx R : x+2x = xx+2o.(c) C = {x Q : x2 (_ + e)x + _e = 0}.(d) D = {n Z : |2n 1| + |2n + 1| 3}.(e) E = {x Z : x2 2x 8 0}.

For a certain set A, the power set of A is P(A) = {, {0},B}, where B is a set. What is A?

Let A and B be sets. Show that if A 6= B, then P(A) 6= P(B).

For S = {3,2, . . . , 2}, list the elements of the set A = {n S : 0 < |n| < 2}.

For S = {1, 2, 3}, give an example of two sets A and B such that |B| = |A| + 1, where A Sand B 6 S.

Give an example of three sets A,B and C such that A B, A / C and B C = .

Determine the power set P(A) for A = {1, 3, 5}.

Let A be the set of students in a discrete mathematics course in your university and B the setof students majoring in computer science in your university. Describe the students in each ofthe following sets.(a) A B. (b) A B. (c) A B. (d) B A.(e) A B. (f) A B. (g) A B.

Let C be the set of professors in a mathematics department who taught a calculus class and Dthe set of professors in a mathematics department who taught a discrete mathematics course.Describe each of the following sets in terms of C and D.(a) the set of professors in the mathematics department who taught both a calculus courseand a discrete mathematics course.(b) the set of professors in the mathematics department who taught either a calculus courseor a discrete mathematics course.(c) the set of professors in the mathematics department who taught neither a calculus coursenor a discrete mathematics course.(d) the set of professors in the mathematics department who taught a calculus course butnot a discrete mathematics course. (e) the set of professors in the mathematics department who taught a discrete mathematicscourse but not a calculus course.1

For A = {1, 2}, B = {1, 0, 1} and the universal set U = {2,1, 0, 1, 2}, determine(a) A B. (b) A B. (c) A B. (d) B. (e) A B.1

For a set S, let P_(S) be the set of all proper nonempty subsets of S. Find P_(S) for each ofthe following. (a) S = {x, y}. (b) S = {1}.1

For S = {{1}, {1, 2}, {1, 2, 3}}, find the power set P(S).1

For the set S = {1, 2, . . . , 6}, give an example of a partition P of S such that |P| = 3.1

Determine whether each of the following statements is true or false.(a) 0 {}. (b) 0 {}. (c) {0}. (d) {0}.(e) {0} {0}. (f) {0} {0}. (g) {} {}. (h) {} {{}}.1

Give examples of two sets A and B such that(a) A B and A B.(b) A B and A B = .(c) A B, A 6 B and A B 6= .1

Give examples of three sets A, B and C such that(a) A B 6 C.(b) A B, B C and A C = .(c) A B, A B and A 6 C.(d) A B, A 6 B and B C.1

For the universal set U = {1, 2, . . . , 10}, draw a Venn diagram for three sets A, B and Cindicating the locations of the elements of U if A, B and C satisfy all of the conditions (a)-(h):(a) A B C = {5},(b) A B C = {10},(c) A (B C) = {3},(d) B (A C) = {4},(e) C (A B) = {2},(f) (B C) A = {1, 2, 4, 9},(g) (A C) B = {2, 3, 7, 8},(h) (A B) C = {3, 4, 6}.1

Let S = {1, 2, . . . , 10}. Give an example of(a) a set A of cardinality 2 such that A P(S).(b) a set B of cardinality 2 such that B P(S).(c) a partition P of S such that |P| = 2.1

Let S = {a, b, c, d, e, f, g}. Give an example of(a) a set A such that A P(S) with |A| = 2.(b) a set B such that B P(S) with |B| = (c) a partition P of S with |P| = 4.2

Let Ai = {1, 2, . . . , i} for 1 i 1 Determine the following.(a)10\i=1Ai. (b)10[i=1Ai.2

For the universal set U = {1, 2, . . . , 8}, let A = {2, 3, 7}, B = {1, 3, 4, 7} and C = {1, 5, 7, 8}.Determine the following.(a) A B. (b) A B. (c) B C.(d) A B. (e) A B C. (f) |A (B C)|.2

Let S = {1, 3, 5, . . . , 15}. For each of the following sets, describe the set by listing its elements.(a) A = {x S : x 8 S}.(b) B = {x S : x 10}.(c) C = {x S : |x 5| S}.2

Let A = {1, {1}} and B = {1, 2, {2}}. Determine the following.(a) A B. (b) B A. (c) A A. (d) P(A). (e) P(B).2

For A = {1, 2, . . . , 10}, let P be a collection of k nonempty subsets of A for some positiveinteger k. Is the following statement true or false? If every element of A does not belong toexactly k 1 subsets in P, then P is a partition of A.2

Determine whether the following statements are true or false.(a) {x} {{x}}. (b) {(x, y)} {x, y}.(c) {x, {x}} {{x}, {{x}}}. (d) {x, {y}} {x, {y}}.(e) {(x, {x})} {x, {x}}. (f) x {{x}, {{x}}}.(g) {({x}, {y})} {{x}, {y}}. (h) {x} {x, {x, {x}}}.2

(a) For A = {1, 2, 3}, give an example of three subsets A1,A2 and A3 of A such that |AiAj | =|i j| for every pair i, j of distinct integers with 1 i, j 3.(b) For B = {1, 2, 3, 4, 5, 6}, give an example of four subsets B1,B2, B3 and B4 of B suchthat |Bi Bj | = |i j| for every pair i, j of distinct integers with 1 i, j 4.2

(a) Let A = {1, 2, 3}. Give an example of three distinct subsets A1,A2 and A3 of A such thatP = {A1 A2,A1 A3,A2 A3} is a partition of A.(b) Let B = {1, 2, 3, 4, 5, 6}. Give an example of four distinct subsets B1,B2, B3 and B4 ofB such that P = {B1 B2,B1 B3,B1 B4,B2 B3,B2 B4,B3 B4} is a partitionof B.