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Sets and relations Reading: Chapter 5 (72-93) from the text book

Sets and relations Reading: Chapter 5 (72-93) from the text book. Sets. We’ll look briefly at the main ideas of sets Our intention is to introduce terminology & notation that will be useful later The term set means a collection of items The items are called the elements of the set

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Sets and relations Reading: Chapter 5 (72-93) from the text book

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  1. Sets and relationsReading: Chapter 5 (72-93) from the text book

  2. Sets • We’ll look briefly at the main ideas of sets • Our intention is to introduce terminology & notation that will be useful later • The term setmeans a collection of items • The items are called the elementsof the set • A set can be described in 2 ways – 1. in enumerated form (i.e. as a list) 2. in predicate form (i.e. using a property that defines the elements of the set)

  3. Examples of Enumerated Sets • the set of summer months is {June, July, August} note the use of braces(‘curly brackets’) for sets • the set of positive even numbers less than 10 is {2, 4, 6, 8} • the set of positive even numbers less than 100 is {2, 4, 6, 8, …, 98} – an ellipsis (3 dots) is used if there is a clear pattern • the set of positive even numbers is {2, 4, 6, 8, …}

  4. Examples of Sets in predicate form • The set of positive even numbers less than 100 can be written in predicate form as {x: x is even and 0 < x < 100} • This definition is read as ‘the set of all x such that x is even and 0 < x < 100’ • Sets are usually denoted by capital letters e.g.A = {2, 4, 6, 8},B = {x: x is a pos. even no.} • The symbols ∈ and ∉ mean ‘is an element of’ and ‘is not an element of’, respectively • e.g.6 ∈ A, 120 ∈ B, 7 ∉ A

  5. Symbols for Special Sets • Special symbols are used for certain sets of nos: N= set of natural numbers = {1, 2, 3, 4, …} J= set of integers = {…, –3, –2, –1, 0, 1, 2, 3,…} Q= set of rationalnos = {x: x = m/n for some integersm andn withn ≠ 0} R= set of all real nos ∅ = the null set or empty set. It has no elements, & may be written as { } or even as {x: x = x + 1} (any predicate that is always false can be used)

  6. The Universal Set • The universal set, denoted by U, contains all elements that could be under discussion in a particular situation • U changes according to circumstances • e.g. If we’re dealing with months of the year, U = {January, February, March, …, December} If we’re dealing with numbers, U might be R(the set of all real nos)

  7. Subsets, Set Operations andVenn Diagrams • If A& B are sets so that every element of Bis an element of A, Bis a subsetof A(written B ⊆ A) • e.g.A = {1,2,3,4}, B = {1,3,4}, C = {4,5,6}. Then B ⊆ A, but C is not a subset of A. • In a picture:

  8. Venn Diagrams • A picture such as in the previous slide is called a Venn diagram • Venn diagrams were introduced by John Venn, who used them in his book Symbolic Logic (1881) to illustrate principles of logic • Venn diagrams are easy to use for 2 or 3 sets. • For more than 3 sets, the diagrams become quite complicated and are not so easy to use.

  9. Properties of Sets • Recall: If every element of Bis an element of A, Bis a subsetof A, written as B ⊆ A • Thus, for any set A, it is true that A ⊆ A • Also, for any set A, it is true that ∅ ⊆ A i.e.we can’t find an element of ∅ which isn’t in A • Two sets Aand Bare equalif A ⊆ B and B ⊆ A Thus 2 sets are equal if they have the same elts • So the orderof listing elts is immaterial e.g. {1, 2, 3} = {2, 1, 3} – & there’s no reason to list an elt more than once – e.g.{1, 2, 1} = {1, 2}

  10. Set Operations • The intersectionof 2 sets A and B is A ∩ B = {x: x ∈ A and x ∈ B} • 2 sets A and B are disjointif A ∩ B = ∅ (i.e. if the sets have no elements in common) • The unionof 2 sets A and B is A ∪ B = {x: x ∈ A or x ∈ B} where ‘or’ means the inclusive‘or’ • The complementof a set A consists of all the elements of the universal set that are not in A. i.e. = {x : x ∈U and x ∉ A}

  11. Set Operations • The differenceof 2 sets A and B is A – B = {x: x ∈ A and x ∉ B} A – B is shaded in red • Note that A − B = A ∩ • This can be shown using the defns of set operations, or by using Venn diagrams

  12. Example Suppose E = {a, b, c, d, e, f, g, h, i, j}, A = {a, b, c, d, e, f, g}, B = {b, d, f, i, j}, C = {a, c, f, j}. Find: (i) A ∪ C (ii) A ∩ B (iii) A∩ C (iv) (B ∩ A) ∪ C (v) A ∪(C ∩ B)

  13. Laws of Sets

  14. Verifying and Using the Laws of Sets • All the laws of sets can be verified • Another way of verifying the laws is to use Venn diagrams • Example: Use Venn diagrams to illustrate the 2nd de Morgan’s law for sets • The laws of sets can be used to simplify a given set (just as we will use the laws of Algebra to simplify a given algebraic expression) • Example: Use laws of sets to simplify

  15. The Power Set • Suppose A = {a, b}. The subsets of A are ∅, {a}, {b} & {a, b} • The set of these subsets is called the powersetof A, denoted by P (A) i.e. P(A) = {∅, {a}, {b}, {a, b}} • Note that P(A) is a set whose elements are themselves sets – i.e. it is a set of sets • Also note that A has 2 elements, & P(A) has 4 elements • Exercise: If A = {a, b, c}, write down P(A)

  16. Cardinality of the Power Set • The cardinalityof a finite set Ais the no. of elements in the set, written as | A | • Example: If A = {a, b, c}, then | A | = 3 • Observe that A has 3 elements, & P(A) has 8 elements • This leads to the general observation: If A has n elements, then P(A) has 2nelements i.e. if | A | = n, then |P(A) | = 2n • Then a set with nelements has 2nsubsets

  17. Ordered Pairs • When dealing with sets, the ordering of elements in the set is immaterial – e.g.{2, 1, 4} = {1, 4, 2} • Sometimes, though, orderdoes matter e.g.: (i) a list of place-getters in a race, or a list of football teams in order of leader position; (ii) an ordered string of characters such as a tax file no., password, credit card PIN or car reg. no. • An ordered pair is a pair of elements in a particular order, written as (a, b)

  18. Ordered n-tuples • Thus the ordered pair (3, 5) is differentto (5, 3) • Note the use of parentheses(‘round brackets’), and not braces (‘curly brackets’) as for sets • If we have nelements, an ordered n-tuple is a list of the n elements in a particular order – it is written as (x1, x2, x3,…, xn) • Since order is important, the only way for (x1, x2, x3,…, xn) = (y1, y2, y3,…, yn) is if the 1st elements are the same (i.e. x1 = y1), the 2nd elements are the same (i.e. x2 = y2), and so on • So (1, 4, 5) ≠ (1, 5, 4) (but {1, 4, 5} = {1, 5, 4})

  19. The Cartesian Product of 2 Sets • The Cartesian product of 2 sets Aand Bis A × B = {(x, y): x ∈ A and y ∈ B} i.e. It is the set of all ordered pairs, where the first element is from A& the second element is from B • e.g. If A is the set of digits 0-9, & B is the set of letters a-z, then (3, t) is in A × B, but (m, 7) is not in A × B – although it is in B × A • e.g. If A = {1, 2, 3} & B = {p, q}, then A × B = {(x, y): x ∈ A and y ∈ B} = {(1, p), (1, q), (2, p), (2, q), (3, p), (3, q)}

  20. The Cartesian Product of n Sets • The Cartesian product of n sets A1, A2,…, Anis A1 × A2 ×… × An= {(x1, x2, x3,…, xn): x1 ∈ A1, x2 ∈ A2, …, xn∈ An} • i.e. It is the set of all ordered n-tuples, where the 1st elt is from A1, the 2nd elt is from A2,etc • e.g. A car reg. no. such as KCT454 can be regarded as an ordered 6-tuple (K, C, T, 4, 5,4). • If L is the set of all letters, & D is the set of all decimal digits, then the set of all possible car registration nos is L × L × L × D × D × D

  21. Cartesian Product of a Set with Itself • The set A × A ×… × A (n times) is written as An • e.g. If Ris the set of real nos, then R2 is the set of all ordered pairs (x, y),where x& y are real nos – geometrically, R2 is the 2-dimensional plane • Similarly, think of R3 as all points in 3-dim space

  22. Cartesian Product of a Set with Itself • e.g.{0, 1}2 = {(0, 0), (0, 1), (1, 0), (1, 1)} • e.g. The elements of {0, 1}nare ordered n-tuples in which each element is 0 or 1 – so a typical element of {0, 1}6is (0, 1, 1, 1, 0, 1) • Think of {0, 1}nas the set of all strings of nbits • Note that L × L × L × D × D × D = L3 × D3

  23. Computer Repn of Sets • To enable computers to handle sets, assume the elements of the universal set U are listed in a definite order. • Then, if |U| = n and Ais a set, Ais represented by a string of nbits b1b2b3…bn. • Here biis 1 if the ith elt of U is in A, and biis 0 if the ith elt of U is not in A.

  24. Computer Repn of Sets • Example: Suppose U = {a, b, c, d, e, f, g}.Find: (a) the representation of {d, f, a, g} as a bit string (b) the set represented by the bit string 0111011 • For sets defined w.r.t. the sameuniversal set, the operations of intersection, union & complement can be carried out directly on the bit strings, without having to convert to the original sets.

  25. Computer Repn of Sets • The bit string of A ∩ B has a 1 if the bit strings of A & B both have a 1, & otherwise has a 0 • This process is termed a bitwiseand operation • The bit string of A ∪ B has a 0 if the bit strings of A & B both have a 0, & otherwise has a 1 (this is a bitwise oroperation)

  26. Set Operations Using Bit Strings • The bit string for the complement of Ais obtained from that of Aby simply replacing 0 with 1, and 1 with 0 (a bitwise notoperation) • Example: Suppose the bit strings of A& B are A: 0110110101, B: 1111001001 Find the bit strings of A ∩ B, A ∪ B &

  27. Relations • A binary relation occurs when we say something about a property of an object relativeto another object of the same type • Example: The statement ‘Ali is taller than Yasir’ illustrates a relation • The word ‘binary’ refers to the fact that twoobjects are compared – in future, we’ll omit this word and refer to just a ‘relation’

  28. Examples which Illustrate Relations • Examples of statements from everyday life which illustrate relations: ‘Ali is the husband of Alia’ ‘Nadir is the sister of Nuha’ ‘Australia has a smaller population than China’ ‘Discrete Maths is a prerequisite for Encryption and Network Security’

  29. More Examples which Illustrate Relations • Examples of statements from mathematics which illustrate relations: ‘12 is greater than 4’ ‘{a} is a subset of {a, b, c}’ ‘20 is divisible by 4’ ‘Line L1 is parallel to line L2’

  30. Comments on the Examples • In each of the examples, a statement is made about a pair of objects of the same type. • The orderof the objects is often important – e.g. it is true that ‘Australia has a smaller population than China’, but it is not true that ‘China has a smaller population than Australia’ • Thus relations involve 2 objects of the same type (i.e. from the same set), where orderis important

  31. Definitions of a Relation • Informal Defn:A relationcan be thought of as a statement about ordered pairs (x, y) that are in A × A, where Ais some set. • This is the basic idea of a relation, although the formal definition looks a little different. • Formal Defn:A (binary) relation on a set Ais a subsetRof A × A. We say that x& yare related iff (x, y) ∈ R.

  32. Example • Consider the relation ‘is greater than’ on the set A = {3, 5, 6, 8}. • For any (x, y) ∈ A × A, either xis greater than y, and then x is related to y or xis not greater than y, and then x is notrelated to y. • The set of the ordered pairs (x, y) ∈ A × A, for which x is related to y is given by: R = {(5, 3), (6, 3), (8, 3), (6, 3), (6, 5), (8, 6)}

  33. Notation for a Relation • In the previous example, we can state that x& yare related by writing (x, y) ∈ R. • In practice, this is often written as xRy(read this as ‘x is related to y’). • For the previous example, we can write ‘x > y’ instead of xRy to mean that xis related to y

  34. Graphical Repn of a Relation • Example:The reln ‘>’ on the set A = {3, 5, 6, 8} can be depicted using a graph. The elements of Aare represented by dots, & if xis related to y, an arrow is drawn from x to y. The result is called a directed graph.

  35. Matrix Repn of a Relation • A relation can also be represented by a matrix (plural ‘matrices’) called the relation matrix. • The entry in row x& column yis T if xis related to y, and is F otherwise. • e.g. For ‘>’ on {3, 5, 6, 8}, the relation matrix is given by

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