Discrete Maths Objective to introduce predicate logic (also called the predicate calculus) , Semester 2, Predicate Logic 1

Overview 1. Motivation 2. Predicate Logic 3. Quantifiers 4. From English to Logic 5. Domain Affects Translation 6. Translation Examples 7. More Information 2

1. Motivation Two sentences: “All men are mortal.” “Socrates is a man.” Is Socrates mortal? These sentences and question cannot be represented in propositional logic. We need a logic that talks about objects (things), their properties, and their relations. 3

2. Predicate Logic Based on prop. logic. The new features: Variables: x, y, z Predicates: P(x), M(x, y), etc. a bit like functions Two quantifiers ( ,  to be covered in a few slides) Predicates can be used to represent properties and relations. 4

A Predicate as a Property Let P(x) mean “x > 0” for integers. Then: P(- 3 ) is falseP( 0 ) is false P( 3 ) is trueP(5) is true We can draw properties as sets, and their objects (things) as elements in the sets: 5 U (domain) is the integers P

A Predicate as a Relation Let “x + y = z” be R(x, y, z) and U be the integers: R( 2,-1, 5 )is false R( 3,4,7 )is true R(x, 3, z)invalid (no unbound variables allowed) Let “x - y = z” be Q(x, y, z), with U as the integers: Q( 2,-1,3 )is true Q( 3,4,7 )is false Q(x, 3, z)invalid 6

Drawing a Relation There's no standard set way of drawing relations, because relations are more powerful than sets. One possible drawing for example 1: 7 U is the integers R... -2—1—5... 3—4—7

Compound Expressions All the operators from prop logic can be used in predicate logic (i.e. , , ,  ) e.g. let P(x) be “x > 0” Then: P( 3 ) ∨ P(-1) is true P( 3 ) ∧ P(-1) is false P( 3 ) → P(1) is true P( 3 ) → P(-1) is false Invalid, since unbound variables are not allowed: P( 3 ) ∧ P(y) P( x ) → P(y) 8

3. Quantifiers We use quantifiers to 'bind' variables in expressions: Two quantifiers: Universal Quantifier, “For all,” symbol:  English: every, all Existential Quantifier, “There exists,” symbol:  English: some, at least one 9 continued

3.1. Universal Quantifier  x P(x) is read as “For all x, P(x)” or “For every x, P(x)” Examples: If P(x) means “x > 0” and U is the integers, then  x P(x) is false If P(x) means “x > 0” and U+ is the positive integers, then  x P(x) is true If P(x) means “x is even ” and U is the integers, then  x P(x) is false 10

P(x) means “x > 0” and U+ is the positive integers, then  x P(x) is true 11  x as a Set Diagram U+ is the positive integers P  x means that all the values in U+ must be inside P; outside P is empty

3.2. Existential Quantifier  x P(x) is read as “For some x, P(x)”, or “There is an x such that P(x),” or “For at least one x, P(x).” Examples: If P(x) means “x > 0” and U is the integers, then  x P(x) is true. If P(x) means “x < 0” and U + is the positive integers, then  x P(x) is false If P(x) means “x is even ” and U is the integers, then  x P(x) is true. 12

 x as a Set Drawing If P(x) means “x > 0” and U is the integers, then  x P(x) is true. e.g. 13 U is the integers P  x means that some of the values in U must be inside P; outside P does not need to be empty

3.3. De Morgan’s Laws for Quantifiers The rules for negating quantifiers are: The table shows that: 14

as Set Diagrams 15 P U not  P U  x P(X) means every value is inside P  x P(X) means not every value is inside P this means there are some values outside P  x P(X) means there are some values inside P  x  P(X) means there are some values outside P the same

4. From English to Logic The translation of English to predicate logic is difficult because we now have quantifiers (  and  ). There are special translations for sentences using: 1. all-are 2. all-and-are 3. some-are 4. no-are 5. some-are-not 6. not-all-are 7. all-are-not 8. only-are 16 these two are the most important to memorize these two are the most important to memorize variations of 4.3 variations of 4.1

4.1. All-are: All p's are q's All   ; are   "All ps are q's" becomes  x (p(x)  q(x)) e.g. "All students are hard-working" becomes  x (s(x)  hw(x)) The common mistake is to translate all-are into  and  "All ps are q's" is NOT  x (p(x)  q(x)) Logically this is "too strong" 17  Memorize this one

All-are as Set Diagrams Why is  and  too strong? Consider non-students. Can non-students be hard-working?  and  allows this, but  and  does not. we use  and  because it is closest to the English meaning 18 U S hw U S  x (s(x)  hw(x))  x (s(x)  hw(x))  "All students are hard-working"

4.2. All-and-are "All men and women are human" becomes  x ( m(x)  w(x)  h(x)) Why not translate to:  x ( m(x)  w(x)  h(x)) The  is "too strong" 19 

Why "too strong"? Consider the meaning of m(x)  w(x) and m(x)  w(x) 20  mw  mw  These are people who are both men and women, which is too small a group. All men and women are...

4.3. Some-are: Some p's are q's Some   ; are   "Some ps are q's" becomes  x (p(x)  q(x)) e.g. "Some women are tall" becomes  x (w(x)  t(x)) Why not:  x (w(x)  t(x)) Logically this is "too weak" 21  Memorize this one

Why too weak? The  and  is too weak since it allows a non-women to be tall which does not 'fit' the English meaning. 22 U w t U wt  x (w(x)  t(x))  x (w(x)  t(x))  Some women are tall

4.4. No-are: No p's are q's "No" = 0 = not (1 or more) =   x "No p's are q's" becomes   x (p(x)  q(x))   x  (p(x)  q(x))   x (  p(x)   q(x))   x (p(x)   q(x)) or   x (  q(x)   p(x)) // reverse the p and q terms   x (q(x)   p(x)) translation + an outer not 4.3 translation + an outer not

4.5 Some-are-not Some p's are not q's becomes  x (p(x)   q(x)) translation + an inner not 4.3 translation + an inner not

4.6. Not-all-are Not all p's are q's becomes  x (p(x)  q(x))   x  (  p(x)  q(x))   x (p(x)   q(x))  "Some p's are not q's" translation + an outer not 4.1 translation + an outer not same as 4.5!

4.7. All-not All p's are not q's becomes  x (p(x)   q(x)) This is the same as 4.4. "No p's are q's" translation + an inner not

4.8. Only-are "Only p's are q's" can be drawn as "subset": Another way of saying this is "All q's are p's" which becomes:  x (q(x)  p(x)) The diagram for this is not the same shape as the one above because it also includes p's which are not q's: 27 U q p U q p

5. Domain Affects Translation “Every student in this class knows Java.” Decide on the domain U. Solution 1 : If U is all students in this class, let J(x) be “x knows Java”. Translate as  x J(x) Solution 2 : If U is all people, let S(x) be “x is a student in this class”. Translate as  x (S(x)→ J(x)) 28 U is 'class students' J U is 'all people' s j This one is better since U is more general. This one is better since U is more general.

6. Translation Examples 1. “Some student in this class has visited Mexico.” Solution: Let M(x) be “x has visited Mexico” and S(x) be “x is a student in this class,” and U be all people.  x (S(x) ∧ M(x)) 2. “Every student in this class has visited Canada or Mexico.” Solution: Let C(x) be “x has visited Canada.”  x (S(x)→ (M(x) ∨ C(x))) 29 Translation 4.3 Translation 4.1

U = {fleegles, snurds, thingamabobs} F(x): x is a fleegle S(x): x is a snurd T(x): x is a thingamabob Translate “Everything is a fleegle” Solution:  x F(x) 30 continued

U = {fleegles, snurds, thingamabobs} F(x): x is a fleegle S(x): x is a snurd T(x): x is a thingamabob “Nothing is a snurd.” Solution: ¬  x S(x) 31 continued Nothing == zero == not (1 or more) == ¬  x Nothing == zero == not (1 or more) == ¬  x

U = {fleegles, snurds, thingamabobs} F(x): x is a fleegle S(x): x is a snurd T(x): x is a thingamabob “All fleegles are snurds.” Solution:  x (F(x)→ S(x)) 32 continued Translation 4.1

U = {fleegles, snurds, thingamabobs} F(x): x is a fleegle S(x): x is a snurd T(x): x is a thingamabob “Some fleegles are thingamabobs.” Solution:  x (F(x) ∧ T(x)) 33 continued Translation 4.3

U = {fleegles, snurds, thingamabobs} F(x): x is a fleegle S(x): x is a snurd T(x): x is a thingamabob “No snurd is a thingamabob.” Solution: ¬  x (S(x) ∧ T(x)) 34 continued Translation 4.4

U = {fleegles, snurds, thingamabobs} F(x): x is a fleegle S(x): x is a snurd T(x): x is a thingamabob “If any fleegle is a snurd then it is also a thingamabob.” Solution:  x ((F(x) ∧ S(x))→ T(x)) 35 "any" suggests  and , but since it is the condition of if-then, just use  "any" suggests  and , but since it is the condition of if-then, just use 

7. More Information See the "Translation Tips" file on the course website goes into more detail about English translations Discrete Mathematics and its Applications Kenneth H. Rosen McGraw Hill, 2007, 7th edition chapter 1, section