CSC 141 Discrete Mathematics Dr. Corina Sas and Ms. Nelly Bencomo

1. CSC 141 Discrete Mathematics Dr. Corina Sas and Ms. Nelly Bencomo c.sas@lancaster.ac.uk; nelly@comp.lancs.ac.uk Computing Department Discrete Mathematics

2. Course details • weeks 1-10 • 10 x 50 min lectures • Material • http://info.comp.lancs.ac.uk/year1/notes/csc141 • coursework • written work (relevant to exam) • recommended resource • http://www.cs.odu.edu/~toida/nerzic/content/web_course.html Discrete Mathematics

3. Syllabus • Sets • Relations • Functions • Recursion • Logic • Boolean • propositional • predicate logic Discrete Mathematics

4. Overview • Discrete Maths • Sets • Defining sets • Set operations • Subsets • Universal and power set Discrete Mathematics

5. Objectives • Understanding the relevance of discrete maths for computer science • Understanding the basic ideas about sets • Facility with basic ideas about sets Discrete Mathematics

6. Discrete maths • What • discrete objects • Why • formal specification • infinity or indefiniteness • reusability Discrete Mathematics

7. Sets and membership • Set = collection of objects • in a set there are no duplicates • a set is Unordered • example set: A = {1, 2, 3, 4, 5, 6, 7} • 1 in set A: • 1 belongs to the set A • 1 is an element/object/member of the set A • Write this: • 1 A • 8 A Discrete Mathematics

8. Defining sets • Listing all its members • writing down all the elements • small, finite sets • A = {a, b, {a, b}, c} • Listing a property that its members must satisfy • {x | 0 < x < 8} • every integer that is greater than 0 and less than 8 • expression on the left does not have to be a variable: • {x-2 | 2 < x < 10} • infinite sets • {x | x > 0} Discrete Mathematics

9. Defining sets • Procedure (program) to generate the members of a set. • a procedure to generate an infinite set is a never ending procedure • EXERCISES. Formally specify the following sets: • all (+ve) even numbers • all (+ve) odd numbers • all (+ve) numbers exactly divisible by 3 • all (+ve) numbers that do not divide exactly by 8 Discrete Mathematics

10. Exercise • Procedure: • Set =  • i = 1 • while i >0 do • put i*3 into Set • i = i+1 • endwhile Exercises – answers {2x | x > 0} {2x + 1 | x ≥ 0} {3x | x > 0} {8x + y | x ≥ 0, 1 ≤ y ≤ 7} Write similar procedure to generate the other sets Discrete Mathematics

11. (all +ve even numbers) Set =  i = 1 while i ≥ 1 do put i * 2 into Set i := i + 1 endwhile (all +ve odd numbers) Set =  i = 0 while i ≥ 0 do put 2 * i + 1 into Set i := i + 1 endwhile (all +ve numbers not exactly divisible by 8) Set =  i = 0 while i ≥ 0 do for j = 1 to 7 do put i + j into Set endfor i = i + 8 endwhile Answers Discrete Mathematics

12. Set Operations • UNION (written ) • takes all of the elements from two sets, and makes a new set containing those elements (with no duplicate elements) • A B = { x | x A x B } • Example: If A = {1, 2, 3} and B = {4, 5} ,  then A B = {1, 2, 3, 4, 5} . • Example: If A = {1, 2, 3} and B = {1, 2, 4, 5} ,  then AB = {1, 2, 3, 4, 5} . • INTERSECTION (written ) • forms a new set from two sets, consisting of all elements that are in BOTH of the original sets • A B = { x | x A  x B } • Example: If A = {1, 2, 3} and B = {1, 2, 4, 5} ,  then A B = {1, 2} . • Example: If A = {1, 2, 3} and B = {4, 5} ,  then A B =  Discrete Mathematics

13. Set Operations • DIFFERENCE (written – or /) • forms a new set from two sets, consisting of all elements from the first set that are not in the second • A- B = { x | x A x B } • Example: If A = {1, 2, 3} and B = {1, 2, 4, 5} ,  then A- B = {3}. • Example: If A = {1, 2, 3} and B = {4, 5} ,  then A- B = {1, 2, 3} . • Each of the basic set operations: • is infix operator • i.e. it is written in between its arguments, as is, for example “+” in arithmetic • takes two sets as its arguments Discrete Mathematics

14. Union • Examples: • {a, b, c}  {b, c, d} = {a, b, c, d} • {a, b, c}  = {a, b, c} •  {a, b, c} = {a, b, c} • {2x | x > 0}  {2x + 1 | x ≥ 0} = {x | x > 0} • NOTE that for all sets A, B, C: • A  B = B  A (commutative law) • A  = A (identity law) • A  A = A (idempotent law) • (A  B)  C = A  (B  C) (associative law) • Checkpoint: “prove” the above to yourself Discrete Mathematics

15. Intersection • Examples • {a, b, c}  {b, c, d} = {b, c} • {a, b, c}  =  •  {a, b, c} =  • {2x | x > 0}  {3x | x > 0} = {6x | x > 0} • {2x | x > 0}  {2x + 1 | x ≥ 0 } =  • NOTE that for all sets A, B, C: • A  B = B  A (commutative law) • A  =  (domination law) • A  A = A (idempotent law) • (A  B)  C = A  (B  C) (associative law) Discrete Mathematics

16. Set difference • Examples: • {a, b, c} - {b, c, d} = {a} • {b, c, d} – {a, b, c} = {d} • {a, b, c} – {} = {a, b, c} • {a, b, c} – {d, e, f} = {a, b, c} • {x | x > 0} – {2x + 1 | x ≥ 0} = {2x | x > 0} • NOTE: for all sets, A • A –  = A •  – A =  • A – A =  Discrete Mathematics

17. Cartesian product • An ordered pair is a pair of objects with an order associated with them. If objects are represented by x and y, then we write the ordered pair as <x, y>. • Two ordered pairs <a, b> and <c, d> are equal if and only if a = c and b = d. For example the ordered pair <1, 2> is not equal to the ordered pair <2, 1>. • The set of all ordered pairs <a, b>, where a is an element of A and b is an element of B, is called the Cartesian product of A and B and is denoted by A x B. • Example 1: Let A = {1, 2, 3} and B = {a, b}. Then A x B = {<1, a>, <1, b>, <2, a>, <2, b>, <3, a>, <3, b>} . • Example 2: For the same A and B as in Example 1, B x A = {<a, 1>, <a, 2>, <a, 3>, <b, 1>, <b, 2>, <b, 3>} . In general, A x B≠B x A unless A =  , B = or A = B. Note that A x = x A =  because there is no element in to form ordered pairs with elements of A. Discrete Mathematics

18. Set operations - Exercises • if A = {a, b, c, y}, B = {a, b, c , d, e} and C = {x, y} evaluate: • A  (B  C) • (A  B)  C • C – A • (A – B) – C • A – (B – C) • (A  C)  B • A  (C  B) Discrete Mathematics

19. Answers • A • {y} • {x} •  • {y} • {a, b, c , d, e, y} • A Discrete Mathematics

20. Subsets • A is a subset of another set, B, means that all members of the set A are also members of the set B. Notation: A  B • we say “A is a subset of B”, or “B is a superset of A” or “A is contained in B” or “B contains A” • Examples: • {a, b, c}  {a, b, c, d, e} • {2x | x > 1}  {x | x > 0} • If A  B, and the set B also contains elements that the set A does not, we say A is a proper subset of B, and we write: A  B • Checkpoint: can we correctly use  instead of  in the two examples above? Discrete Mathematics

21. Subset Exercises • if A = {a, b, c, d, e, f}, B = {a, b, e}, C = {c, d}, and D = {d, f, g} say which of the following are true statements: • B  B • B  B • B  A • C  A • (B  C)  A • D  A • (D  C)  A • (D  C)  A Discrete Mathematics

22. Answers • true (of any set) • false (of any set) • true • true • true • false • true • false Discrete Mathematics

23. Universal sets • we often consider sets in terms of them all being subsets of a so called universal set or universe • for example, all the sets of numbers we have considered so far have been subsets of the set of all positive integers • the so-called natural numbers, sometimes written N • this leads to the notion of the complement of a set • the complement is the difference between the universe and a given set • e.g. with a universe of N, the complement of {2x | x > 1} is {2x + 1 | x > 0} • the complement of a set A is usually written Ā • we’ll write comp(A) Discrete Mathematics

24. Power sets • Universal sets - examples • suppose the universe, U = {a, b, c, d, e, f, g}, A = {a, b, c} and B = {b, c, d, e} • comp(A) = {d, e, f, g} • comp(B) = {a, f, g} • comp (A)  comp(B) = {a, d, e, f, g} • comp(A  B) = comp({b, c}) = {a, d, e, f, g} • comp(A)  comp(B) = {f, g} • comp (A  B) = comp({a, b, c, d, e}) = {f, g} • Power Sets - the set of all subsets of a set A is called the power set of A and denoted by   2A  or  (A) . • For example for A = {1, 2},   • (A) = {, {1}, {2}, {1, 2} } Discrete Mathematics