Download Study Guide Unit Test2 with Sample Problems

CmSc 175 Discrete Mathematics
Study Guide for Unit Exam 2
Predicate Logic, Valid and Invalid Arguments in Predicate Logic
Proofs, Mathematical Induction,
Set, Boolean Algebra
1. Be able to translate universally and existentially quantified statements in predicate
logic and find their negation
2. Be able to recognize valid and invalid arguments in predicate logic, determine the
inference rule applied and the types of errors.
3. Know how to prove statements using direct proofs and mathematical induction.
4. Know set identities
5. Know the formal definitions of set operations: union, intersection, difference and
complement.
6. Be able to compute union, complement, intersection of discrete sets and of sets of
real numbers
7. Know how to simplify set expressions and to show that two expressions are
equivalent using set identities and the formal definitions of set operations.
8. Know how to find the powerset of a given set, the Cartesian product of two sets,
and to determine if a given set of subsets of some set is a partition or not.
9. Know the axioms of Boolean Algebra
10. Be able to prove statements in Boolean algebra, e.g. x + x = x
Sample problems with solutions are given below. See also problems in homework
assignments and quizzes
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1. Be able to translate universally and existentially quantified statements in predicate
logic and find their negation
Men are mortal
x, man(x)  mortal(x)
Negation: x, man(x)  ~mortal(x)
Some men are immortal
Some cheaters sit at the back row
x, cheater(x)  sit_in_the_back_row (x)
Negation: x, cheater(x)  sit_in_the_back_row (x) (x)
No cheaters sit in the back row
Negations of quantified statements
Babies cry.
Some babies don’t cry
No textbook is cheap
Some textbooks are cheap
Some students study in the library
No students study in the library
Some students don’t take introductory classes
All students take introductory classes
2. Be able to recognize valid and invalid arguments in predicate logic, determine the
inference rule applied and the types of errors.
All honest people pay their taxes.
Darth is not honest
Therefore Darth does not pay his taxes
Invalid, Inverse error
Senior students must take a capstone class
Anna is a senior student
Therefore Anna must take a capstone class.
Valid, Modus Ponens
All teachers occasionally make mistakes.
No gods make mistakes
Therefore no teachers are gods
Explanation:
x, teacher(x)  make_mistakes(x)
x, god(x)  ~make_mistakes(x)
x, make_mistakes (x)  ~ god (x)
Valid Hypothetical Syllogism.
(1)
(2)
(2’ contrapositive of (2))
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Now combine (1) and (2’) with HS, we get:
x, teacher(x)  ~ god (x)
All teachers occasionally make mistakes.
Peter makes mistakes
Therefore Peter is a teacher
Invalid converse error
All teachers occasionally make mistakes.
Peter does not make mistakes
Therefore Peter is not a teacher
Valid Modus Tollens
3. Know how to prove statements using direct proofs and mathematical induction.
3.1. Prove that the sum of two odd numbers is an even number.
Proof:
Let x and y be two odd numbers.
By the definition of odd numbers, there is some p integer such that x = 2p -1, and
there is some q integer such that y = 2q – 1.
x + y = 2p – 1 + 2q – 1 = 2p + 2q – 2 = 2(p + q – 1)
Let s = p + q – 1
Then, x + y = 2s
x + y has the representation of an even number, therefore it is even.
3.2. Let S(n) = 1 + 2 + 3 + 4 + …. + n, n = 1, 2, 3, ….
Prove by mathematical induction the statement
P(n): S(n) = n(n+1)/2 for all integers n  1
a. Base step
Show that P(1) is true.
P(1): S(1) = 1*(1 + 1)/2 = 1
By the definition of S(n), S(1) = 1
Therefore P(1) is true
b. Inductive step
Show that P(k) P(k+1) is true
P(k): S(k) = k(k+1)/2
P(k+1): S(k+1) = (k+1)(k+2)/2
Assume that P(k) is true. We have to show that P(k+1) is also true.
S(k+1) = S(k) + (k+1)
= k(k+1)/2 + (k+1) =
= k(k+1)/2 + 2(k+1)/2
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= ((k+1)(k+2))/2
Therefore P(k+1) is true, therefore P(k) P(k+1) is true.
By the principle of math induction,
P(n): S(n) = n(n+1)/2 is true for all integers n  1
4. Know set identities
See Lesson 11
5. Know the formal definitions of set operations: union, intersection, difference and
complement.
A  B = {x | (x є A )  (x є B)}
A  B = {x | (x є A ) V (x є B)}
A – B = {x | (x є A ) V (x  B)}
~A = {x | (x  A ) }
6. Be able to compute union, complement, intersection of discrete sets and of sets of
real numbers
Let
A = {x | -1 < x < 6 V 10  x 15}
B = {x | 4 < x  12}
A  B = { x | 4 < x < 6 V 10  x  12}
A  B = { x | -1 < x 15 }
~A = { x | x  -1 V 6  x < 10 V 15 < x}
~B = { x | x  4 V 12 < x}
~A  ~B = {x | x  -1 V 15 < x}
~A  ~B = {x | x  4 V 6  x < 10 V 12 < x}
7. Know how to simplify set expressions and to show that two expressions are
equivalent using set identities and the formal definitions of set operations.
Show that
(A  B)  B = B
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a. Proof using set identities
(A  B)  B =
=
=
=
=
=
=
=
(A  B)  ( B  U) by identity law
(A  B)  ( B  (A  ~A)) by complementation law
(A  B)  (B  A)  (B  ~A) by distributive law
(B  A)  (B  A)  (B  ~A) by commutative laws
(B  A)  (B  ~A) by idempotent laws (i.e. A  A = A)
B  (A  ~A) by distributive law
BU
by complementation law
B
by identity law
b. Proof using the formal definitions
(A  B)  B = {x | ((x є A )  (x є B)) V (x є B)} =
= {x | (x є B)} by the absorption laws in propositional logic
(P V (P Λ Q) ≡ P
Here P is x є B, Q is x є A
8. Know how to find the power set of a given set, the Cartesian product of two sets,
and to determine if a given set of subsets of some set is a partition or not.
a. Power set
A = {1,2} P (A) = {, {1}, {2}, {1,2}}
Note: The power set of a set with n elements has 2n elements
A =  P (A) = {}
A = {P (A) = {, {}, {}, {, } }
b. Cartesian products
A = {1,2}, B = {a, b, c}, C = {3}
A x B = {(1,a), (1,b), (1,c), (2,a), (2,b), (2,c)}
B x A = {(a,1), (a,2), (b,1), (b,2), (c,1), (c,2)}
A x C = {(1,3), (2,3)}
The number of pairs in the Cartesian product is the product of the number of
elements of each set.
c. Partitions
A = {1,2,3,4,5}, Which of the following sets are not partitions? Explain
{{1,2},{3,5},{4}} Partition
{{1,2,3},{4,5}} Partition
{1,2},{4,5}} Not a partition, element 3 is missing
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{{1,2},{2,3,4,5}} Not a partition, element 2 is repeated, i.e. the sets {1,2} and
{2,3,4,5} are not disjoint
9. Know the axioms of Boolean Algebra
See Lesson 13
10. Be able to prove statements in Boolean algebra, e.g. x + x = x
Prove x = x + x
x
=x+0
= x + x . ~x
= (x + x) . (x + ~x)
= (x + x) . 1
=x+x
by identity axiom
by complements axiom
by distributive axiom
by complements axiom
by identity axiom
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