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 ⊂ U
(b) B ⊆ A
(c) ø ⊂ 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 ⊂ 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 ∈ A; A ⊂ B
(b) x ∈ A; A and B are disjoint
(c) x ∈ A; x ∉ B; B ⊂ A
(d) x ∈ A; x ∈ B; A is not a subset of B; B is not a subset of A
(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 ∩ B, A ∪ C, A′, B′, B ∩ A′, B ∩ C′, A – B, A Δ B
(c) Complete the statement using a single symbol: C - B = ... .
2
True or false?
(a) | ø | = 1
(b) | { x, x } | = 2
(c) | U ∩ ø | = 0
3
What can you say about two sets P and Q if:
(a) P ∩ Q′ = ø
(b) P ∪ Q = P?
Question 4
4
Make six copies of the Venn diagram shown alongside, and then shade the areas represented by:
(a) A ′ ∪ B
(b) A ∩ B ′
(c) (A ∩ B) ′
(d) A ′ ∪ B ′
(e) (A ∪ B) ′
(f) A ′ ∩ B ′
5
Identify the sets represented by each of the shaded areas below, using the set notation symbols ∩, ∪ and ′ only:
(a)
(b)
(c)
(d)
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 ∩, ∪ and ′.
(b) Again using one of your answers to question 5, write a definition of A Δ B using only symbols from the list ∩, ∪ and ′. (There are two possibilities here – see if you can find them both!)
(d) What could you say about two sets A and B if A × B = B × 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) ∈ ... }
4
It is believed that, for any sets A, B and C, A × (B ∩ C) = (A × B) ∩ (A × C).
(Note that, if this is true, it says that × is distributive over ∩.)
Copy and complete the two Cartesian diagrams shown below – one for the expression on each side of the equation – to investigate this.