Introduction to Sets and Logic

1
Introduction to Sets and Logic

• Why do we need to know a bit of set theory?
• Set theory allows us to describe and classify
  entities
• So it allows us to describe and group the data
  that we may need to use for testing
• Set theory allows us to operate on groups of
  entities
• So it allows us to operate on groupings of test
  data
• Set theory allows us to extend the study to
  relations and functions
• So it allows us to relate test data or test
  scenarios and expected results
• Why do we need to know a bit about logic?
• Propositional logic allows us to manipulate
  statements and truth values of the statements
  predicate logic allows us to introduce
  quantifiers such as there exists or all
• So it allows us to describe and combine test
  conditions and deduce validity of test coverage
  or validity of test results.

Set theory and logic, while useful for all
aspects of testing, it is most useful with
functional testing (black box testing)
2
What is a set ?

• A set is a collection or a group of something.
• For example a set of items in a classroom may
  be represented as follows.
• Classrooom_items chair1, chair2, desk1,
  student 1, student2, desk2, desk3, chair3,
  instructor, whiteboard, computer
• Student_ages 18, 17, 23, 19
• A word of caution a set can not have repetition.
• If you had two students of the same age, then you
  may be tempted to say
• Student_ages 17, 18, 18, 23
• but in set theory you can only say
  
• Student_ages 17, 18, 23

incorrect
correct
3
Different Ways to Represent a Set

• A) Listing the elements that belong to the set,
  just as shown earlier
• my_lucky_ numbers 3, 106, 7689, -4
• B) Providing a decision rule that defines the set
• possible_male_ages age 0 lt age 200
• C) Combinations of or redefinition of sets
• students_in_class boy_students U girl_students

There is a special set called the empty set which
contains no element. It is represented with the
symbol, Ø. For example the set,
possible_male_age age age -3, is an
empty set. So this set is Ø.
4
Common Set Operations

• Union union of set A and set B is the
  combination of sets A and B, represented as A U
  B.
• If A apple, orange, grapes and B shoe,
  tie, shirt, then AUB apple, orange, grapes,
  shoe, tie, shirt
• Intersection intersection of set A and B is the
  set whose elements are composed of only the
  common elements from set A and set B, represented
  as A n B.
• If A and B are as defined above, then AnB
  Ø
• If A apple, grape, orange and B
  tomato, grape, lettuce, then AnB grape
• Complement complement of a set A is the set that
  has all the elements that are not in A,
  represented as A or A
• If A numbers numbers gt 0 , then A
  numbers numbers 0

5
Some more Set Operations

• Minus set A minus set B is the set which
  contains all elements in A that is not in B,
  represented as (A B). this set is sometimes
  known as relative complement of B with respect
  to A.
• If A 2, 34, 67, 98, -3, -54, -300 and B
  integers integers 0, then (A B)
  2,34,67,98
• Symmetric Difference the symmetric difference of
  sets A and B, represented as A B, is a set
  that is defined as the result of operations, A
  U B A n B
• If A apple, orange, grape and B pear,
  grape, orange, then A B apple, pear

6
Specification vs Implementation vs Test Cases
Specification
Implementation
Actual
Expected
2
3
1
5
4
6
7
Look at this earlier Venn Diagram
Tested
The area represented by number 2 may be
represented as Expected nActual Tested
Test Cases
7
A few More Definitions

• Set A is a subset of set B, written A B, if
  every element in A is also an element in B.
• Set A is a proper subset of set B, written A
  B, if it is a subset of B and there exists at
  least one element in B that is not in A (or B A
  ? Ø).
• Set A and set B are equal, written A B, if
  every element in A is in B and vise versa.

8
Set Partition

• A partitioning of a set is a division of a set
  into groups. This is a useful concept for many
  areas where we want to take samples from groups
  formed based on some criteria.
• Example performing an opinion survey of
  teenagers on clothes. We may partition the survey
  by
• Age groups - 13 to 15 , 16 to 17, 18 to
  19
• Gender - male , female
• Types of schools - Catholic, Jewish,
  Baptist, non-religious public
• This concept is important when we have to design
  test cases based on picking the input data.
• Formally, a dividing of a set X into subset x1,
  x2, - - - xn is called a partitioning of X if the
  following is true
• a) x1 U x2 U - - - U xn X
• b) for any xi and xk , where i ?
  k , xi n xk Ø

For later Discuss - partitioning the outputs of
the triangle problem for testing
9
Some Rules about sets

• A U Ø A
• A n Ø Ø
• (A) A
• Commutative rules A U B B U A A n B B n A
• Associative rules A U (B U C) (A U B) U C
• A n (B n C)
  (A n B) n C
• Distributive rules A n (B U C) (A n B) U (A n
  C)
• A U (B n C) (A U
  B) n (A U C)
• DeMorgans rules (A U B) A n B
• (A n B)
  A U B

10
Informally - Relation

• A Relation is, as the English word says, a
  connection between two sets.
• Given a set A Tom, Sam, Sally, Jane and a set
  B (Joe, Mary, Abe, then a relation such as
  parent of may be defined between these set B and
  A. If Joe is the father of Sam and Mary is the
  mother of Tom, then the relation, B parent of
  A, is true with some of the elements.
• For the above sets of A and B, there may or may
  not be any meaningful relation.

11
Relationship between Two Sets
SWE_students
Good_Programs
written by
The arrows represent the relationship, written
by. Then one can see that 3 of the
Good_Programs are written by 2 of the SWE
students.
12
Relation (more formally)

• Given two sets A and B, the cross-product of A x
  B is the set composed of paired items, where
  first element of the pair is from A and the
  second element of the pair is from B.
• Example A 3. 4, 8 and B mod_x, mod_y
  then cross-product of A x B (3, mod_x), (4,
  mod_x), (8,mod_x),
• (3,
  mod_y), (4, mod_y) , (8, mod_y)
• Given a set A and a set B, a relation, R, is a
  subset of cross-product of A and B. So R A x
  B .
• Example A tom, mary and B abu, kathy,
  A x B (tom, abu), (tom, kathy), (mary, abu),
  (mary, kathy) .
• If kathy is the sister of tom, then a
  relation, R, called brother of is a subset of A
  x B where brother of (tom, kathy)

13
Some Properties of Relation

• A relation, R1, defined over a set A, is said to
  be reflexive if for every a in set A, (a,a) is in
  R1.
• Example If A is set of positive integers and R1
  is the relation, then R1 is reflexive because
  every positive integer itself. But if R1 were
  defined as gt, then R1 is not reflexive.
• A relation, R2, is said to be symmetric if for
  every (a,b) in R2, (b,a) is also inR2.
• Example if for (a,b) in R2 means a is bs
  classmate, then b is also as classmate. Thus
  (b,a) is also in R2. But if R2 is defined as
  (a,b) mean a is bs father, the R2 is not
  symmetric.
• A relation, R3, is said to be transitive if for
  every (a.b) and (b,c) in R3 implies that (a,c) is
  in R3.
• Example If R3 is defined as gt, then (7,4) and
  (4,2) in R3 implies that (7,2) is also in R3
  because 7 gt 2.
• Example R3 defined as mother of relation is
  not transitive

There is a relation called equivalence relation
which is reflexive, symmetric, and transitive. We
will use the partitioning of test data into
subsets that are each considered to be an
equivalence class and pick a test case from each
class.
14
Function

• Function is a special type of relation.
• A relation, RF, is a function if for every (x,y)
  in RF and (x,z) in RF implies that y z.

RF is a function
RF is a non-function
Is the inverse, RF-1, a function?
Is the inverse, RF-1, a function?
15
Simple Logic Propositional Calculus

• A proposition is a statement.
• A proposition, just like a statement, carries the
  value of True or False, but not both.
• Example Let X be the proposition, Howard is a
  student in SWE 3643. Then X is either True or
  False.
• Example Let Y be the proposition, It will rain
  tomorrow. Is Y True or False ?

Note that it will rain tomorrow is not a
proposition that can be handled by propositional
calculus because its truth vale can not
determined until tomorrow.
16
Basic Operators of Propositional Calculus

• Let P and Q be propositions, then the truth
  values of 5 logical operators (NOT,AND, OR,
  Exclusive OR, Implies) is shown in the following
  truth table.

p
q
P ? q
q
P v q
P q

P ? q
T
T
F
T
T
F
T
T
T
F
F
T
T
F
F
T
F
F
T
T
T
F
T
T
F
F
F
F
Legend is NOT ? is logical AND v is
logical OR is logical Exclusive OR ? is
logical Implies.

17
Logical Implies

• All the truth operations are quite intuitive,
  except for implies, also known as if ---then
• P implies Q intuitively says that when P is true
  then Q must be true
• It is also intuitive that when P is true then Q
  can not be false.
• What happens when P is NOT True?
• Propositional Calculus defines this situation as
  True regardless of the truth value of Q.
• Sometimes P implies Q, P ?Q, is stated as
  (P AND Q ) OR (NOT P) .

Look at the previous truth table for (P? Q) OR (
P)
18
Some Logical Expression Rules

• Simple ones
• P ? True P
• P v False P
• P v True True
• P ? False False
• ( P) P
• P ? Q Q ? P
• P v Q Q v P
• Associative (P v Q) v R P v (Q v R) (
  P ? Q) ? R P ? (Q ? R)
• Distributive P ? (Q v R) (P ? Q) v (P ? R)
• P v (Q ? R) (P v Q)
  ? (P v R)
• Demorgans Law (P ? Q) P v Q
• (P v
  Q) P ? Q

Use the truth table to prove to yourself one of
the DeMorgans Law