Foundations of Discrete MathematicsChapter 2By Dr. Dalia M. Gil, Ph.D.

SetsSet: A collection of things called elements or members.

The set of natural numbers N consists of the numbers 1, 2,... Their members are all positive.

SetsThe set of integers Z consists of the natural numbers (1, 2, ), their negatives (, -3, -2, , 2, 3, , and 0. Zero (0) is an integer, but not a natural number.

Ways to Describe Sets {egg1, egg2}

{x}

N = {1, 2, 3, }

Z = {, -3, -2, -1, 0, 1, 2, 3, } This set has two elements This set has one element The set of natural numbers The set of integer numbers

Describing a Set with a Builder Notation{ x | x has certain properties }such thatThe set of x such that x has certain properties.We read:

Describing a Set with a Builder Notation{ some expression | the expression has certain properties }{n | n is an odd integer, n > 0}Example: the set of odd natural numbers.such that

Describing a Set with a Builder Notation{2k 1 | k = 1, 2, 3, }Example: the set of odd natural numbers.K belongs to N{2k 1 | k N}or

Describing a Set with a Builder Notationm Z m is an integerThe symbol denoting set membership0 N negates the meaning of

Describing a Set with a Builder NotationQ = {m/n| m, n Z, n 0}The set of rational numbers QThe members of Q are fractions, which are ratios of integers with nonzero denominators.Examples , -2/17, 5(=5/1)

Describing a Set with a Builder NotationThe set of irrational numbers.The members of irrational set cannot be written in the form m/n with m and n both integers.The decimal expansions of the irrational numbers neither terminate or repeat. Examples 2, 317, e, , ln 5

Describing a Set with a Builder NotationThe set of complex numbers C.The members of complex set have the form a + biwhere a and b are real numbers, i2 = -1C = {a + bi | a, b R, i2 = -1}

Describing a Set with a Builder NotationA set can be an element of another set{{a, b}, c} is a set with two elements, {a, b} and c.

Equality of SetSets A and B are equal, and we write A = B, if and only if A and B contain the same elements or neither set contains any element. {1, 2, 1} = {1, 2} = {2, 1}

{1/2, 2/4, -3/-6}, /2} = {1/2}

{t|t = r s, r, s {0, 1, 2}} = {-2, -1, 0, 1, 2}

The Empty SetThe empty set is a set that contains no elements.P = {n N | 5n = 2}

S = {n N | n2 + 1 = 0}

The set small of people less than 1 millimeter.

These sets are all equal since none of them contains any elements.

SubsetsA set A is a subset of a set B (A B), if and only if every element of A is an element of B.If A B but A B, then A is called a proper subset of B and we write A B

SubsetsA B A is contained in B A is a subset of BB A B is superset of A

Examples of Subsets{a, b} {a, b, c}{a,b} is a subset of {a,b,c}{a, b} {a, b, c}{a,b} is a proper subset of {a,b,c}

Examples of Subsets{a, b} {a, b, {a, b}}{a,b} is a subset of {a,b,{a,b}}{a, b} {a, b, {a, b}}{a,b} belongs to {a,b,{a,b}}{a,b} is an element of {a,b,{a,b}}

Examples of SubsetsN Z Q R CThe set of natural numbers is a proper subset of the set of integer numbers. The set of integer numbers is a proper subset of the set of rational numbers.

Examples of SubsetsN Z Q R CThe set of rational numbers is a proper subset of the set of real numbers. The set of real numbers is a proper subset of the set of complex numbers.

Subsets. Proposition 1For any set A, A A and A ProofIf a A, then a A, so A A If A is false, then there must exist some x such that x A. This an absurdity since there is no x

Subsets. Proposition 2If A and B are sets, then A = B if and only if A B and B A Proof() If A = B, then A is a subset of B and B is a subset of A. () If A is a subset of B and B is a subset of A, then A = B.

Subsets. Proposition 2a b membershipa b, a is an element of the set b.a b, a is a set each of whose elements is also in the set b.a b subset

The Power SetThe power set of a set A, denoted P(A), is the set of all subsets of A: P(A) = {B | B A}

Examples of The Power SetIf A = {a}, then P(A) = {, {a}} If A = {a, b}, then P(A) = {, {a}, {b}, {a, b}} P ({a, b, c}) = {, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}

Union and IntersectionThe union of sets A and B, A B, is the set of elements in A or in B (or in both).The intersection of sets A and B, A B, is the set of elements that belongs to both A and B.

Examples: Union and IntersectionIf A = {a, b, c} and b = {a, x, y, b} A B = { a, b, c, x, y}

A B = {a, b}A {} = {a, b, c, }

B {} =

Examples: Union and IntersectionFor any set A,

A = A and

A =

Union and IntersectionThe union and intersection of sets are associative operations.(A1 A2) A3 = A1 (A2 A3)For any three sets A1, A2, A3 , the expression A1 A2 A3 is unambiguous.

Union and IntersectionThe union of n sets A1 A2 A3 An is written ni=1AiRepresents the set of elements that belong to one or more of the sets Ai

Union and IntersectionThe intersection the sets A1, A2, An is written ni=1AiRepresents the set of elements which belong to all of the sets Ai

Union and IntersectionA = {1, 2, 3, 4}B = {3, 4, 5, 6}C = {2, 3, 5, 7}B C= {2, 3, 4, 5, 6, 7}A (B C)= {2, 3, 4}

Union and IntersectionA = {1, 2, 3, 4}B = {3, 4, 5, 6}C = {2, 3, 5, 7}(A B) C= {2, 3, 4, 5, 7}A B = {3, 4}

Union and Intersection(A B) C= {2, 3, 4, 5, 7}A (B C)= {2, 3, 4}A (B C) (A B) C

Union and Intersection(A B) = {3, 4}A (B C)= {2, 3, 4}A (B C) = (A B) (A C)(A C)= {2, 3}(A B) (A C)= {2, 3, 4}

Union and IntersectionLet A, B, and C be sets. VerifyA (B C) = (A B) (A C)Solution using proposition 2:If A and B are sets, then A = B if and only if A B and B A

Union and IntersectionTo show A, B, and C be sets. VerifyA (B C) (A B) (A C)Let x A (B C) Then x is in A and also in B C, Since x B C, either x B or x C. This suggests two cases

Union and IntersectionCase 1: x BIn this case, is in A as well as in B, so its in A B Case 2: x CIn this case, is in A as well as in B, so its in A C

Union and IntersectionWe have shown that either x A B or x A CBy definition of union, x (A B) (A C)

Union and IntersectionWe must show that A (B C) (A B) (A C)

Let x (A B) (A C)Then either x (A B) or x (A C)Thus, x is in both A and B or in both A and C. In either case x A. Also x is in either B or C; thus x B C

Union and IntersectionWe must show that A (B C) (A B) (A C)

Let x (A B) (A C)Then either x (A B) or x (A C)So x is in both A and in B C ; that is x A ( B C).

Set Difference The set difference of sets A and B (A\ B), is the set of those elements of A that are not in B.The complement of a set A is the set Ac = U \ A, where U is some universal set made clear by the context.

Examples: Set Difference{a, b, c} \ {a, b} = {c}{a, b, c} \ {a, x} = {b, c}{a, b, } \ = {a, b}

Examples: Set Difference{a, b, } \ {} = {a, b, }If A is the set {Monday, Tuesday, Wednesday, Thursday, Friday}, so

Ac = {Saturday, Sunday}

Examples: Set DifferenceA \ B = A Bc and (Ac)c= A Example (suppose U = Z): If A = {x Z | x2 > 0}, then Ac ={0}

(Ac)c = {0}c = {x Z | x 0} = A

Interval NotationIf a and b are real numbers with a < b, then[a, b] = {x R | a x b} closed(a, b) = {x R | a < x < b} open

Interval NotationIf a and b are real numbers with a < b, then(a, b] = {x R | a < x b} half open[a, b) = {x R | a x < b} half open

The Laws of De Morgan(A B)c = Ac Bc(A B)c = Ac Bc

x (A B)c (x A B)Prove that (A B)c = Ac Bc for any set A, B, and C.Let A be the statement x A and B be the statement x B

x (A B)c (x A B) x Ac BcProve that (A B)c = Ac Bc for any set A, B, and C. (A or B) x Ac and x Bc A and BDefinition of unionRule for negating orDefinition of intersection

Symmetric DifferenceThe symmetric difference of two sets A and B is the set A B of the elements that are in A or in B, but not in both.A B = (A B) \ ( A B)A B = (A \ B) ( A \ B)

Examples: Symmetric Difference{a, b, c} {x, y, a} = { b, c, x, y}{a, b, c} = {a, b, c}{a, b, c} {} = {a, b, c, }

Use a Venn diagram to illustrate that for any three sets A, B, and C, (A B) C = A (B C)A B = {1, 2, 5, 6}(A B) C = {1, 3, 6, 7}C = {2, 3, 5, 7}

Use a Venn diagram to illustrate that for any three sets A, B, and C, (A B) C = A (B C)B C = {2, 7, 4, 6}A (B C) = {1, 3, 6, 7}A = {1, 2, 3, 4}

The Cartesian Product of SetsIf A and B are sets, the Cartesian product (the direct product) of A and B is the setA x B = {(a, b) | a A, b B} A cross B for A x B

The Cartesian Product of SetsA1 x A2 x An =When all the sets are equal to the same set A, A x A x x A is written An.= {a1, a2, an)| ai Ai for i=1, 2, , n}

The Cartesian Product of SetsThe elements of A x B are called ordered pairs because the order is important:(a, b) (b, a)

(a, b)= (b, a) if a = b

The Cartesian Product of SetsThe first coordinate(a, b) The second coordinateThe elements of An are called n-tuples

The Cartesian Product of SetsElements of A x B are equal if and only if they have the same first coordinates and the same second coordinates(a1, b1) = (a2, b2) if and only if a1 = a2 and b1 = b2

Example: The Cartesian Product of SetsLet A = {a, b} and B = {x, y, z}. Then B x A = {(x, a), (x, b), (y, a), (y, b), (z, a), (z,b)}A x B B x AA x B = {(a, x), (a, y), (a, z), (b, x), (b, y), (b,z)}

Example: The Cartesian Product of SetsLet A, B, and C be sets. Prove that A x (B C) (A x B) (A x C)Solution:Prove that any element in A x (B C) is in (A x B) (A x C).

Let A, B, and C be sets. Prove that A x (B C) (A x B) (A x C)The elements in A x (B C) are ordered pairs (x, y).

where x, the first coordinate, is in A and y, the second coordinate is in (B C).

y is in either B or C.

If y is in B, then, since x coordinate is in A, (x, y) A x B.

If y is in C, then, since x coordinate is in A, (x, y) A x C.

Thus, (x, y) is either in A x B or in A x C; thus, (x, y) is in (A x B) (A x C), which is what we wanted to show. Let A, B, and C be sets. Prove that A x (B C) (A x B) (A x C)

An element of (A x B) (A x C) is either in A x B or in A x C.

The elements in (A x B) (A x C) are ordered pairs (x, y).

(x, y) A x B or (x, y) A x C. Let A, B, and C be sets. Prove that (A x B) (A x C) A x (B C)

If y is in B, then, since x coordinate is in A, (x, y) A x B.

If y is in C, then, since x coordinate is in A, (x, y) A x C.

In either case, x is in A and y is either in B or in C; so x A and y B C. Therefore, (x, y) A x B C. Let A, B, and C be sets. Prove that (A x B) (A x C) A x (B C)

In this example, we show that (A x B) (A x C) A x (B C)

In the previous example, we showed that A x (B C) (A x B) (A x C)

Conclusion: A x (B C) = (A x B) (A x C)Let A, B, and C be sets. Prove that (A x B) (A x C) A x (B C)

() Suppose that A x B = B x A is true.

Suppose x A. Since B , we can find some y B. Thus, (x, y) A x B.

(x, y) B x A. So x B, giving us A B. Similarly, B A. Conclusion A = B.

() If A = B is true, thenA x B = A x A = B x ALet A and B be nonempty sets. Prove that A x B = B x A A = B

If A = and B is nonempty set,

Then A x B = = B x A, but A B.

So A x B = B x A does not mean A = B in the case B = . Let A and B be nonempty sets. Prove that A x B = B x A A = B. Is this true if A = ?

If A and B are sets. A binary relation from A to B is a subset of A x B.

A binary relation on A is a subset of A x A.

The empty set and the entire Cartesian product A x B are always binary relations from A to B. Binary Relations

Let A be the set of students who are registered at VCC during the Summer 2008 semester.

Let B be the set {database, discrete mathematics, English}

R = {(a, b) | a A is enrolled in a course in subject b B} Examples of Binary Relations

Let A is the set of surnames of people listed in the Sprint telephone directory.

Then R = {(a, n) | a appears on page n} is a binary relation from A to the set N of natural numbers.Examples of Binary Relations

{(a, b} | a, b N, a/b is an integer} is a binary relation on N

{(a, b) | a, b N, a b = 2} is a binary relation on N

{(x, y) | y = x2} is a binary relation on R . Examples of Binary Relations

A binary relation R on a set A is reflexive if and only if (a, a) R for all a A. Binary RelationsA reflexive relation must contain all pairs of the form (a, a).

{(x, y} R2 | x y } is a reflexive relation on R since x x for any x R.

{(a, b) N2 | a/b N} is a reflexive relation on N since a/a is an integer, 1, for any a N. Examples of Reflexive Binary Relations

R = {(x, y} R2 | x2 + y2 > 0} is not a reflexive relation on R since (0, 0) R.

Examples of Reflexive Binary Relations

Suppose

R = {(a, b} | a, b Z x Z | a2 = b2}

Criticize and then correct the following proof that R is reflexive:

(a, a) R if a2 = a2. Reflexive Relations

The statement (a, a) R if a2 = a2 is the implication a2 = a2 (a, a) R. For any integer a, we have a2 = a2 and, hence, (a, a) R. Therefore, R is reflexive.

Reflexive Relations

A binary relation R on a set A is symmetric if and only

if a, b A and (a, b) R, then (b, a) RSymmetric Relations

R = {(x, y} R2 | x2 + y2 =1} is a symmetric relation on R since if x2 + y2 =1 then y2 + x2 = 1 too: If (x, y) R, then (y, x) R.

(x, y} Z2 | x - y is even} is a symmetric relation on Z since if x - y is even so is y x.Examples of Symmetric Relations

R = {(x, y} R2 | x y} is not a symmetric relation on R.

For example, (2, 1) R because 22 1, but (2, 1) R because 12 < 2.Examples of Symmetric Relations

R = {(1,1), (1, 2), (1, 4), (2, 1), (2, 2), (3, 2), (3, 3), (4, 4)}Examples of RelationsGiven the set A = {1, 2, 3, 4}R is reflexive, but not symmetric

A binary relation R on a set A is antisymmetric if and only

if a, b A and both (a, b) and (b, a) are in R, then a = b.Antisymmetric Relations

R = {(x, y) R2 | x y} is an antisymmetric relation on R since x y and y x implies y = x; thus (x, y) R and (y, x) R implies x = y.

If S is a set and A = P (S) is the power set of S, then {(X, Y) | X, Y P (S), X Y} is antisymmetric since X Y and Y X implies X = Y.Examples of Antisymmetric Relations

R = {(1, 2), (2, 3), (3, 3), (2, 1)} is not antisymmetric on A ={1, 2, 3} because (1, 2) R and (2, 1) R but 1 2.

Is not symmetric because (2, 3) R but (3, 2) R.

antisymmetric is not the same as not symmetric.Examples of Antisymmetric Relations

Is the relation R = {((x, y), (u, v)) R2 x R2 | x2 + y2 = u2 + v2 }antisymmetric? NO ((1, 2), (2, 1)) R because 12 + 22 = 22 + 12 ((2, 1), (1, 2)) R because 22 + 12 = 12 + 22 Examples of Antisymmetric Relationsbut (1, 2) (2, 1)

A binary relation R on a set A is transitive if and only

if a, b, c A and both (a, b) and (b, c) are in R, then (a, c) R.Transitive Relations

R = {(x, y) R2 | x y} is a transitive relation on R since,

if x y and y z, then x z:

if (x, y) and (y, z) are in R then (x, y) R.

Examples of Transitive Relations

{(a, b) Z x Z | a/b is an integer} is a transitive relation on Z since,

if a/b and b/c are integers, then

so is a/c because a/c = a/b . b/c.Examples of Transitive Relations

R = {(x, y) (x, z), (y, u), (x, u) is a transitive binary relation on the set {x, y, z, u}

there is only one pair of the form (a, b), (b, c) belonging to R (x, y) and (y, u)) and, for this pair,

it is true that (a, c) = (x, u) R. Examples of Transitive Relations

R = {(a, b), (b, a), (a, a)} is not transitive the pairs (b, a) and (a, b), but not the pair (b, b).

{(a, b)| a and b are people and a is an ancestor of b} is a transitive relation since if a is an ancestor of b and b is an ancestor of c, then a is an ancestor of c. Examples of Transitive Relations

An n-ary relation on sets

A1, A2, , An is a subset of A1 x A2 x An

Binary relation is 2-ary relationHigh-Order Relations

An equivalence relation on a set A is a binary relation R on A.

It is reflexive, symmetric, and transitive. Equivalence Relations

Suppose A is the set of all people in the world and

R = {(a, b) A x A | a and b have the same parents}

Show that R is an equivalence relation?Examples of Equivalence Relations

R is reflexive (each person has the same set of parents as himself/herself).

R is symmetric (if a and b have the same parents, then so do b and a),

R is transitive (if a and b have the same parents, and b and c has the same parents, then a and c have the same parents).R = {(a, b) A x A | a and b have the same parents} is reflexive because

If R is a binary relation on a set A, and a, b A, to prove that R is an equivalence, we must prove that R is

reflexive: a ~ a for all a A,Symmetric: if a A and b A and a ~ a, then b ~ a, andTransitive: if a, b, c A and both a ~ b and b ~ c, then a ~ c.Equivalence Relations (~)

Let A be the set of students registered at VCC.

For a, b A, call a and b equivalent if their student numbers have the same first two digits.a ~ a, for ever student a because any number has the same first two digits, so the student numbers of b and a have the same first two digits. Therefore b ~ aExamples of Equivalence Relations

Finally, if a ~ b and b ~ c, then the student numbers of a and b have the same first two digits, and the students numbers of b and c have the same first two digits, so the student numbers of a and c have the same first two digits.

R is reflexive, symmetric, and transitive, R is an equivalence relation on A.Examples of Equivalence Relations

The three most fundamental properties of equality are

reflexive: a = a for all a,Symmetric: if a = b, then b = a; andTransitive: if a = b and b = c A and both a ~ b and b ~ c, then a = c.Equality is an equivalence relation

The groups into an equivalence relation divides the underlying set are equivalence classes.

The equivalence class of an element is the collection of all things related to it.Equality is an equivalence relation

The equivalence class of an element a A is the set a= {x A | x ~ a} for all element to a.

The set all equivalence classes is called the quotient set of A mod ~ and denoted A/~.The Equivalence Class

An equivalence relation is symmetric, so x ~ a or a ~ x in the definition of the equivalence class of .

The set of things related to a is the same as the set of things to which a is related. The Equivalence Class

A/~ is the quotient set of A.

The set of equivalence classes of the equivalence relation ~ on a set A.

is the equivalence class of a; also used to denote the congruence class of an integer a (mod n)A/~ and

Let A be the set of students registered at VCC.

The equivalence relation is formed by the students who are related to a particular student x and those whose student number have the same first two digits as xs student number.Examples of the quotient set

An equivalence class is formed by the students who is the set of all students whose student numbers begin with the same first two digits.

The quotient set is the set of all equivalence classes:A/~ ={class of n| n = 05, 04, 03, 02, 01, 00, 99, 98, }Examples of the quotient set

For (x, y) and (u, v) in R2, define (x, y) ~ (u, v) if x2 + y2 = u2 + v2 .

Prove that ~ defines an equivalence relation on R2 and interpret the equivalence classes geometrically.Example of Equivalence Relations

If (x, y) R2, then x2 + y2 = u2 + v2 , so (x, y) ~ (u, v): The relation is reflexive.

If (x, y) ~ (u, v), then x2 + y2 = u2 + v2 , so u2 + v2 = x2 + y2 and (u, v) ~(x, y): The relation is symmetric.

Example of Equivalence Relations

If (x, y) ~ (u, v) and (u, v) ~(w, z), then x2 + y2 = u2 + v2 and u2 + v2 = w2 + z2 Thus x2 + y2 = u2 + v2 = w2 + z2

Since x2 + y2 = w2 + z2 (x, y) ~ (w, z), so the relation is transitive.Example of Equivalence Relations

The equivalence class of (a, b) is

= {(x, y)| (x, y) ~(a, b)}={(x, y) | x2 + y2 = a2 + b2 }

={(x, y) | x2 + y2 = 12 + 02 = 1}Geometric Interpretation Graph of a circle in the Cartesian plane with center (0, 0) and radius 1.

()Suppose = , x because x ~ x, so x ; thus, x ~ a. () Suppose that x ~ a. Prove that the two sets and are equal.

Suppose y . Then y ~ x and x ~ a, so y ~ a by transitivity. Thus y , Let ~ denote an equivalence relation on a set A. Then, for any x A, x ~ a if an only if =

On the other hand. Suppose y . Then y ~ a. Since a ~ x, we have both y ~ a and a ~ x; therefore, by transitivity, y ~ x. Thus y and . Therefore = Let ~ denote an equivalence relation on a set A. Then, for any x A, x ~ a if an only if =

A partition of a set A is a collection of disjoint nonempty subsets of A whose union is A.

These disjoint sets are called cells ( or blocks).

The cells are said to partition A.A Partition

Canada is partitioned into ten provinces and three territories.

Students are partitioned into groups according to the first two digits of their student numbers.

The human race is partioned into groups by eye color.Examples of Partitions

A deck of playing cards is partitioned into four suits.

If A = {a, b, c, d, e, f, x}, then {{a, b}, {c, d, e}, {f}, {x}} is a partition of A. So is {{a, x}, {b, d, e, f}, {c}}.Examples of Partitions

The equivalence classes associated with an equivalence relation on a set A form a partition of A.Theorem 2.4.6

Given the set {a, b, c, d, e, f, g}

The partition {{a, g}, {b, d, e, f}, {c}} corresponds to the equivalence relation whose equivalence classes are

{a, g}, {b, d, e, f} and {c}Examples of Partitions

The equivalent relation (x ~ y) is represented in the figure by means a cross in row x and column y.

Examples of Partitions

Partial Orders A partial order on a set A is a binary relation that is reflexive, antisymmetric, and transitive.

A partially ordered set, poset for short, is a pair (A, ), where is partial order on the set A.

Partial Orders Writing a b to mean that (a, b) is in the relation, a partial order on A is a binary relation that is

reflexive: a a for all a A,Symmetric: if a, b A, a b and b a, then b = a, andTransitive: if a, b, c A, a b and b c, then a c.

Partial Orders a b (a is less than b) signifies a b, a b, just as we use a < b (a is less than b) to mean a b, a b.

Examples of Partial Orders The binary relation on the real numbers is a partial order because

a a for all a R (reflexive),

a b and b a implies a = b (antisymmetric), and

a b, b c implies a c (transitivity).

Examples of Partial Orders Suppose some alphabet of symbols ordered partially by some relation . For words a = a1a2 an and b = b1b2 bn define a b if a and b are identical

a b in the alphabet at the first position I where the words differ, or

n < m, and ai = bi for i = 1, , nSuppose some alphabet of symbols ordered partially by some relation . For words a = a1a2 an and b = b1b2 bn define a b if a and b are identical

a b in the alphabet at the first position I where the words differ, or

n < m, and ai = bi for i = 1, , n

Partial Orders If (A, ) is a partially ordered set, elements a and b of A are said to be comparable if and only if either a b or b a.The definition of partial order does not require that every pair of elements be comparable, in the following sense.

Examples Partial Orders If X and Y are subsets of a set S, it need not be the case that X Y or Y X

For example {a} and {b, c} are not comparable.

Totally Ordered Set 2. If is a partial order on a set A and every two elements of A are comparable, then is called a total order and the pair (A, ) is called a totally ordered set.

Examples of Totally Ordered Set The real numbers are totally ordered by because, for every pair a, b of real numbers, either a b or b a.The set of sets, { {a}, {b}, [c}, {a, c}} is not totally ordered by since neither {a} {b} nor {b} {a}.

Maximum and Minimum ElementsAn element a of a poset (A, ) is maximum if and only if b a for every b A and minimum if and only if a b for every b A .In the poset (P({a, b, c}), ), is a minimum elementThe set {a, b, c} a maximum element

Maximal and Minimal ElementsAn element a of a poset A is maximal if and only if,If b A and a b, then b = a.and minimal if and only if,If b A and b a, then b = a.

Maximal and Maximum ElementsWhat are the maximum, minimum, maximal, and minimal elements in the poset of the following Hasse diagram.

a and b are maximal.d is both minimum and minimal.There is no maximumMaximal and Maximum Elements

g a, g b, andIf c a and c b for some c A, then c g. Elements a and b can have at most one glb. When this element exists, it is denoted a ^ b (a meet b).A Greatest Lower Bound (glb)Let (A, ) be a poset. An element g is a greatest lower bound (glb) of elements a, b A if and only if

a l, g l, andIf a c, b c for some c A, then l c The lub is unique if it exists. The lub of a and b is denoted a b, a join b if there is such an element.A Least Upper Bound (lub)An element l is a least upper bound (lub) of a and b if

Example of glb and lubIn the poset (R, ), the greatest lower bound (glb) of two real numbers is the smaller of the two and the lower upper bound (lub) the larger.

A poset (A, ) in which every two elements have a greatest lower bound in A and a least upper bound in A is called a latticeA Lattice

In the poset (R, ), the greatest lower bound (glb) of two real numbers is the smaller of the two and the lower upper bound (lub) the larger.Example of a Lattice

Topics coveredSets.

Operations on Sets.Union and IntersectionSet DifferenceSymmetric Difference

Binary Relations

Equivalence Relations.

Partial Order

ReferenceDiscrete Mathematics with Graph Theory, Third Edition, E. Goodaire and Michael Parmenter, Pearson Prentice Hall, 2006. pp 38-71.