Let A and B be sets. Prove or disprove: jA Bj D jA Bj C jB Aj

Let A D f1; 2; 3; 4; 5g and let B D f4; 5; 6; 7g. Please compute: a. A [ B. b. A \ B. c. A B. d. B A. e. A B. f. A B. g. B A. 1

Let A and B be sets with jAj D 10 and jBj D 7. Calculate jA \ Bj C jA [ Bj and justify your answer.

Let A and B be sets with jAj D 10 and jBj D 7. What can we say about jA [ Bj? In particular, find two numbers x and y for which we can be sure that x  jA [ Bj  y and then find specific sets A and B so that jA [ Bj D x and another pair of sets so that jA [ Bj D y. Finally, answer the same question about jA \ Bj (find upper and lower bounds as well as examples to show that your bounds are tight).

a. A line in the plane is a set of points. If `1 and `2 are two different lines, what can we say about j`1 \ `2j? In particular, find all possible values of j`1 \ `2j and interpret them geometrically.b. A circle in the plane is also a set of points. If C1 and C2 are two different circles, what can we say about jC1 \ C2j? Again, interpret your answer geometrically

Prove Theorem 12.3.

Let A and B be sets. Explain why A \ B and A B are disjoint.

Earlier we presented a Venn diagram of the distributive property A [ .B \ C / D .A [ B/ \ .A [ C /. Please give a Venn diagram of the other distributive property, A \ .B [ C / D .A \ B/ [ .A \ C /.

Is a Venn diagram illustration a proof? (This is a philosophical question.)

Suppose A, B, and C are sets with A \ B \ C D ;. Prove or disprove: jA [ B [ Cj D jAj C jBj C jCj.

Suppose A, B, and C are pairwise disjoint sets. Prove or disprove: jA [ B [ Cj D jAj C jBj C jCj.

Let A and B be sets. Prove or disprove: A [ B D A \ B if and only if A D B

Let A and B be sets. Prove or disprove: jA Bj D jAj C jBj jA \ Bj.

Let A and B be sets. Prove or disprove: jA Bj D jA Bj C jB Aj.

Let A be a set. Prove: A ; D A and ; A D ;.

Let A be a set. Prove: A A D ; and A ; D A.

Prove that A  B if and only if A B D ;

Let A and B be nonempty sets. Prove: A B D B A if and only if A D B. Why do we need the condition that A and B are nonempty?

State and prove necessary and sufficient conditions for AB D B A. In other words, create a theorem of the form Let A and B be sets. We have A B D B A if and only if (a condition on A and B). Then prove your result.

Give a standard proof of Proposition 12.12 and illustrate it with a Venn diagram.

In Exercise 11.8 we defined what it means for a region in the plane to be convex, namely, a region R is convex provided given any two points in R, the line segment joining those points is entirely contained in R. Prove or disprove the following assertions: a. The union of two convex regions is convex. b. The intersection of two convex regions is convex.

True or False: For each of the following statements, determine whether the statement is true or false and then prove your assertion. That is, for each true statement, please supply a proof, and for each false statement, present a counterexample (with explanation). In the following, A, B, and C denote sets. a. A .B C / D .A B/ C. b. .A B/ C D .A C / B. c. .A [ B/ C D .A C / \ .B C /. d. If A D B C, then B D A [ C. e. If B D A [ C, then A D B C. f. jA Bj D jAj jBj. g. .A B/ [ B D A. h. .A [ B/ B D A.

Let A be a set. The complement of A, denoted A, is the set of all objects that are not in A. STOP! This definition needs some amending. Taken literally, the complement of the set f1; 2; 3g includes the number 5, the ordered pair .3; 4/, and the sun, moon, and stars! After all, it says . . . all objects that are not in A. This is not what is intended. When mathematicians speak of set complements, they usually have some overarching set in mind. For example, during a given proof or discussion about the integers, if A is a set containing just integers, A stands for the set containing all integers not in A. If U (for universe) is the set of all objects under consideration and A  U, then the complement of A is the set of all objects in U that are not in A. In other words, A D U A. Thus ; D U. Prove the following about set complements. Here the letters A, B, and C denote subsets of a universe set U. a. A D B if and only if A D B. b. A D A. c. A [ B [ C D A \ B \ C.The notation A is handy, but it can be ambiguous. Unless it is perfectly clear what the universe set U should be, it is better to use the set difference notation rather than complement notation.

Design a four-set Venn diagram. Notice that the three-set Venn diagram we have been using has eight regions (including the region surrounding the three circles) corresponding to the eight possible memberships an object might have. An object might be in or not in A, in or not in B, and in or not in C. Explain why this gives eight possibilities. Your Venn diagram should show four sets, A, B, C, and D. How many regions should your diagram have? On your Venn diagram, shade in the set A B C D. Note: Your diagram does not have to use circles to demark sets. Indeed, it is impossible to create a Venn diagram for four sets using circles! You need to use other shapes.

Let A, B, and C be sets. Prove that jA [ B [ Cj D jAj C jBj C jCj jA \ Bj jA \ Cj jB \ Cj C jA \ B \ Cj:

There is an intimate connection between set concepts and Boolean algebra concepts.The symbols ^ and _ are pointy versions of \ and [, respectively. This is more than a coincidence. Consider: x 2 A\B () .x 2 A/^.x 2 B/ x 2 A[B () .x 2 A/_.x 2 B/ Find similar connections between the set-theoretic notions of  and with notions from Boolean algebra.

Prove that symmetric difference is a commutative operation; that is, for sets A and B, we have A B D B A.

Prove that symmetric difference is an associative operation; that is, for any sets A, B, and C, we have A .B C / D .A B/ C.

Give a Venn diagram illustration of A .B C / D .A B/ C.

Prove Proposition 12.15.

Let A, B, and C denote sets. Prove the following: a. A .B [ C / D .A B/ [ .A C /. b. A .B \ C / D .A B/ \ .A C /. c. A .B C / D .A B/ .A C /. d. A .B C / D .A B/ .A C /. 13 Combina