Discrete Mathematics/Set theory/Exercises

Set Theory Exercise 1
1
 * Is each of the following a well-defined set? Give brief reasons for each of your answers.


 * (a) The collection of all alphanumeric characters.


 * (b) The collection of all tall people.


 * (c) The collection of all real numbers x for which:
 * 2x – 9 = 16.


 * (d) The collection of all integers x for which:


 * 2x – 9 = 16.


 * (e) The collection of all good tennis players.

2
 * U = {natural numbers}; A = {2, 4, 6, 8, 10}; B = {1, 3, 6, 7, 8}


 * State whether each of the following is true or false:


 * (a)	2 &isin; A


 * (b)	11 &isin; B


 * (c)	4 &notin; B


 * (d)	A &isin; U


 * (e)	A = {even numbers}

3
 * U = R; A = {4, &radic;2, 2/3, -2.5, -5, 33, &radic;9, &pi;}


 * Using the {…} set notation, write the sets of:


 * (a) natural numbers in A


 * (b) integers in A


 * (c) rational numbers in A


 * (d) irrational numbers in A

4
 * True or false?


 * (a) &oslash; = {0}


 * (b) x &isin; { x }


 * (c) &oslash; = { &oslash; }


 * (d) &oslash; &isin; { &oslash; }

5
 * The following sets have been defined using the | notation. Re-write them by listing some of the elements.


 * (a) {p | p is a capital city, p is in Europe}


 * (b) {x | x = 2n - 5, x and n are natural numbers}


 * (c) {y | 2y2 = 50, y is an integer}


 * (d) {z | 3z = n2, z and n are natural numbers}

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Set Theory Exercise 2
1
 * Copy the truth table to the right, and write at the end of each row the number of the corresponding region in Fig. 4 Venn Diagrams.

2
 * If U = {letters of the alphabet}, A = {a, a, a, b, b, a, c}, B = {c, b, a, c} and C = {a, b, c}, what can be said about A, B and C?

3
 * U = {natural numbers}; A = {2, 4, 6, 8, 10}; B = {1, 3, 6, 7, 8}


 * State whether each of the following is true or false:


 * (a) A &sub; U


 * (b) B &sube; A


 * (c) &oslash; &sub; U

4
 * U = {a, b, c, d, e, f, g, h}; P = {c, f}; Q = {a, c, d, e, f, h}; R = {c, d, h}


 * (a) Draw a Venn diagram, showing these sets with all the elements entered into the appropriate regions. If necessary, redraw the diagram to eliminate any empty regions.


 * (b) Which of sets P, Q and R are proper subsets of others? Write your answer(s) using the &sub; symbol.


 * (c) P and R are disjoint sets. True or False?

5
 * Sketch Venn diagrams that show the universal set, U, the sets A and B, and a single element x in each of the following cases:


 * (a) x &isin; A; A &sub; B


 * (b) x &isin; A; A and B are disjoint


 * (c) x &isin; A; x &notin; B; B &sub; A


 * (d) x &isin; A; x &isin; B; A is not a subset of B; B is not a subset of A

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Set Theory Exercise 3
1
 * U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
 * A = {2, 4, 6, 8, 10}
 * B = {1, 3, 6, 7, 8}
 * C = {3, 7}


 * (a) Illustrate the sets U, A, B and C in a Venn diagram, marking all the elements in the appropriate places. (Note: if any region in your diagram does not contain any elements, re-draw the set loops to correct this.)


 * (b) Using your Venn diagram, list the elements in each of the following sets:


 * A &cap; B, A &cup; C,  A&prime;,  B&prime;,  B &cap; A&prime;,  B &cap; C&prime;,  A – B,  A &Delta; B


 * (c)	Complete the statement using a single symbol: C - B = ....

2
 * True or false?


 * (a) | &oslash; | = 1


 * (b) | { x, x } | = 2


 * (c) | U &cap; &oslash; | = 0

3
 * What can you say about two sets P and Q if:


 * (a) P &cap; Q&prime; = &oslash;
 * (b) P &cup; Q = P?



4
 * Make six copies of the Venn diagram shown alongside, and then shade the areas represented by:


 * (a) A &prime; &cup; B


 * (b) A &cap; B &prime;


 * (c) (A &cap; B) &prime;


 * (d) A &prime; &cup; B &prime;


 * (e) (A &cup; B) &prime;


 * (f) A &prime; &cap; B &prime;

5
 * Identify the sets represented by each of the shaded areas below, using the set notation symbols &cap;, &cup; and &prime; only:

6
 * (a) One of the shaded regions in question 5 represents the set A – B. Identify which one it is, and hence write a definition of A – B using only symbols from the list &cap;, &cup; and &prime;.


 * (b) Again using one of your answers to question 5, write a definition of A &Delta; B using only symbols from the list &cap;, &cup; and &prime;. (There are two possibilities here – see if you can find them both!)

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Set Theory Exercise 4
1
 * (a) If A = {1, 2, 3, 4}, write down P(A) by listing its elements. What is the value of | P(A) |?


 * (b) If | A | = 5, what is the value of | P(A) |?


 * (c) If | A | = 10, what is the value of | P(A) |?

2
 * Prove the following identities, stating carefully which of the set laws you are using at each stage of the proof.


 * (a) B &cup; ( &oslash; &cap; A) = B


 * (b) (A &prime; &cap; U) &prime; = A


 * (c) (C &cup;  A) &cap; (B &cup; A) = A &cup; (B &cap; C)


 * (d) (A &cap; B) &cup; (A &cap; B ' ) = A


 * (e) (A &cap; B) &cup; (A &cup; B ' ) &prime; = B


 * (f) A &cap; (A &cup; B) = A

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Set Theory Exercise 5
1
 * X = {a, c} and Y = {a, b, e, f}.


 * Write down the elements of:


 * (a) X &times; Y


 * (b) Y &times; X


 * (c) X2 (= X &times; X)


 * (d) What could you say about two sets A and B if A &times; B = B &times; A?

2
 * A chess board’s 8 rows are labelled 1 to 8, and its 8 columns a to h. Each square of the board is described by the ordered pair (column letter, row number).


 * (a) A knight is positioned at (d, 3). Write down its possible positions after a single move of the knight.


 * (b) If R = {1, 2, ..., 8}, C = {a, b, ..., h}, and P = {coordinates of all squares on the chess board}, use set notation to express P in terms of R and C.


 * (c) A rook is positioned at (g, 2). If T = {2} and G = {g}, express its possible positions after one move of the rook in terms of R, C, T and G.

3
 * In a certain programming language, all variable names have to be 3 characters long. The first character must be a letter from 'a' to 'z'; the others can be letters or digits from 0 to 9.


 * If L = {a, b, c, ..., z}, D = {0, 1, 2, ..., 9}, and V = {permissible variable names}, use a Cartesian product to complete:


 * V = {pqr | (p, q, r) &isin;  ... }

4
 * It is believed that, for any sets A, B and C, A &times; (B &cap; C) = (A &times; B) &cap; (A &times; C).


 * (Note that, if this is true, it says that &times; is distributive over &cap;.)


 * Copy and complete the two Cartesian diagrams shown below – one for the expression on each side of the equation – to investigate this.


 * Do you think that the proposition is true?



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