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… … and now for something and now for something completely different…completely different…

Set TheorySet TheoryActually, you will see that logic Actually, you will see that logic and set theory are very closely and set theory are very closely

related.related.

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Set TheorySet Theory

• Set: Collection of objects (“elements”)Set: Collection of objects (“elements”)

• aaAA “a is an element of A” “a is an element of A” “a is a member of A” “a is a member of A”

• aaAA “a is not an element of A” “a is not an element of A”

• A = {aA = {a11, a, a22, …, a, …, ann} } “A contains…”“A contains…”

• Order of elements is meaninglessOrder of elements is meaningless

• It does not matter how often the same It does not matter how often the same element is listed.element is listed.

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Set EqualitySet Equality

Sets A and B are equal if and only if they Sets A and B are equal if and only if they contain exactly the same elements.contain exactly the same elements.

Examples:Examples:

• A = {9, 2, 7, -3}, B = {7, 9, -3, 2} :A = {9, 2, 7, -3}, B = {7, 9, -3, 2} : A = BA = B

• A = {dog, cat, horse}, A = {dog, cat, horse}, B = {cat, horse, squirrel, dog} : B = {cat, horse, squirrel, dog} : A A B B

• A = {dog, cat, horse}, A = {dog, cat, horse}, B = {cat, horse, dog, dog} : B = {cat, horse, dog, dog} : A = BA = B

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Examples for SetsExamples for Sets

““Standard” Sets:Standard” Sets:

• Natural numbers Natural numbers NN = {0, 1, 2, 3, …} = {0, 1, 2, 3, …}

• Integers Integers ZZ = {…, -2, -1, 0, 1, 2, …} = {…, -2, -1, 0, 1, 2, …}

• Positive Integers Positive Integers ZZ++ = {1, 2, 3, 4, …} = {1, 2, 3, 4, …}

• Real Numbers Real Numbers RR = {47.3, -12, = {47.3, -12, , …}, …}

• Rational Numbers Rational Numbers QQ = {1.5, 2.6, -3.8, 15, = {1.5, 2.6, -3.8, 15, …}…}(correct definition will follow)(correct definition will follow)

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Examples for SetsExamples for Sets

• A = A = “empty set/null set”“empty set/null set”

• A = {z}A = {z} Note: zNote: zA, but z A, but z {z}{z}

• A = {{b, c}, {c, x, d}}A = {{b, c}, {c, x, d}}

• A = {{x, y}} A = {{x, y}} Note: {x, y} Note: {x, y} A, but {x, y} A, but {x, y} {{x, y}} {{x, y}}

• A = {x | P(x)}A = {x | P(x)}“set of all x such that P(x)”“set of all x such that P(x)”

• A = {x | xA = {x | xNN x > 7} = {8, 9, 10, …}x > 7} = {8, 9, 10, …}“set builder notation”“set builder notation”

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Examples for SetsExamples for Sets

We are now able to define the set of rational We are now able to define the set of rational numbers Q:numbers Q:

QQ = {a/b | a = {a/b | aZZ bbZZ++} }

or or

QQ = {a/b | a = {a/b | aZZ bbZ Z bb0} 0}

And how about the set of real numbers R?And how about the set of real numbers R?

RR = {r | r is a real number} = {r | r is a real number}

That is the best we can do.That is the best we can do.

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SubsetsSubsetsA A B B “A is a subset of B” “A is a subset of B”A A B if and only if every element of A is also B if and only if every element of A is also an element of B. an element of B.We can completely formalize this:We can completely formalize this:A A B B x (xx (xA A x xB)B)

Examples:Examples:

A = {3, 9}, B = {5, 9, 1, 3}, A A = {3, 9}, B = {5, 9, 1, 3}, A B ? B ?

truetrue

A = {3, 3, 3, 9}, B = {5, 9, 1, 3}, A A = {3, 3, 3, 9}, B = {5, 9, 1, 3}, A B ? B ?

falsefalse

truetrue

A = {1, 2, 3}, B = {2, 3, 4}, A A = {1, 2, 3}, B = {2, 3, 4}, A B ? B ?

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SubsetsSubsetsUseful rules:Useful rules:• A = B A = B (A (A B) B) (B (B A) A) • (A (A B) B) (B (B C) C) A A C C (see Venn Diagram)(see Venn Diagram)

UU

AABB

CC

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SubsetsSubsets

Useful rules:Useful rules: A for any set A A for any set A • AA A for any set A A for any set A

Proper subsets:Proper subsets:

A A B B “A is a proper subset of B”“A is a proper subset of B”

A A B B x (xx (xA A x xB) B) x (xx (xB B x xA)A)

oror

A A B B x (xx (xA A x xB) B) x (xx (xB B x xA) A)

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Cardinality of SetsCardinality of Sets

If a set S contains n distinct elements, nIf a set S contains n distinct elements, nNN,,we call S a we call S a finite setfinite set with with cardinality ncardinality n..

Examples:Examples:

A = {Mercedes, BMW, Porsche}, |A| = 3A = {Mercedes, BMW, Porsche}, |A| = 3

B = {1, {2, 3}, {4, 5}, 6}B = {1, {2, 3}, {4, 5}, 6} |B| = |B| = 44C = C = |C| = 0|C| = 0

D = { xD = { xN N | x 7000 }| x 7000 } |D| = 7001|D| = 7001

E = { xE = { xN N | x 7000 }| x 7000 } E is infinite!E is infinite!

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The Power SetThe Power Set

P(A) P(A) “power set of A”“power set of A”

P(A) = {B | B P(A) = {B | B A} A} (contains all subsets of A)(contains all subsets of A)

Examples:Examples:

A = {x, y, z}A = {x, y, z}

P(A)P(A) = {= {, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, , {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z}}{x, y, z}}

A = A = P(A) = {P(A) = {}}

Note: |A| = 0, |P(A)| = 1Note: |A| = 0, |P(A)| = 1

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The Power SetThe Power SetCardinality of power sets:Cardinality of power sets:

| P(A) | = 2| P(A) | = 2|A||A|

• Imagine each element in A has an “Imagine each element in A has an “onon//offoff” switch” switch• Each possible switch configuration in A Each possible switch configuration in A

corresponds to one element in 2corresponds to one element in 2AA

zzzzzzzzzzzzzzzzzzyyyyyyyyyyyyyyyyyyxxxxxxxxxxxxxxxxxx8877665544332211AA

• For 3 elements in A, there are For 3 elements in A, there are 22222 = 8 elements in P(A)2 = 8 elements in P(A)

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Cartesian ProductCartesian ProductThe The ordered n-tupleordered n-tuple (a (a11, a, a22, a, a33, …, a, …, ann) is an ) is an

ordered collectionordered collection of objects. of objects.

Two ordered n-tuples (aTwo ordered n-tuples (a11, a, a22, a, a33, …, a, …, ann) and ) and

(b(b11, b, b22, b, b33, …, b, …, bnn) are equal if and only if they ) are equal if and only if they

contain exactly the same elements contain exactly the same elements in the same in the same orderorder, i.e. a, i.e. aii = b = bii for 1 for 1 i i n. n.

The The Cartesian productCartesian product of two sets is defined as: of two sets is defined as:

AAB = {(a, b) | aB = {(a, b) | aA A b bB}B}

Example:Example: A = {x, y}, B = {a, b, c} A = {x, y}, B = {a, b, c}AAB = {(x, a), (x, b), (x, c), (y, a), (y, b), (y, c)}B = {(x, a), (x, b), (x, c), (y, a), (y, b), (y, c)}

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Cartesian ProductCartesian Product

The The Cartesian productCartesian product of two sets is defined of two sets is defined as: Aas: AB = {(a, b) | aB = {(a, b) | aA A b bB}B}

Example:Example:

A = {good, bad}, B = {student, prof}A = {good, bad}, B = {student, prof}

AAB = {B = {(good, student),(good, student), (good, prof),(good, prof), (bad, student),(bad, student), (bad, prof)(bad, prof)}}

(student, good),(student, good), (prof, good),(prof, good), (student, bad),(student, bad), (prof, bad)(prof, bad)}} BA = BA = {{

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Cartesian ProductCartesian Product

Note that:Note that:

• AA = = • A = A = • For non-empty sets A and B: AFor non-empty sets A and B: AB B A AB B B BAA

• |A|AB| = |A|B| = |A||B||B|

The Cartesian product of The Cartesian product of two or more setstwo or more sets is is defined as:defined as:

AA11AA22……AAnn = {(a = {(a11, a, a22, …, a, …, ann) | a) | aiiAAii for 1 for 1 i i

n}n}

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Set OperationsSet Operations

Union: AUnion: AB = {x | xB = {x | xA A x xB}B}

Example:Example: A = {a, b}, B = {b, c, d} A = {a, b}, B = {b, c, d}

AAB = {a, b, c, d} B = {a, b, c, d}

Intersection: AIntersection: AB = {x | xB = {x | xA A x xB}B}

Example:Example: A = {a, b}, B = {b, c, d} A = {a, b}, B = {b, c, d}

AAB = {b}B = {b}

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Set OperationsSet Operations

Two sets are called Two sets are called disjointdisjoint if their intersection if their intersection is empty, that is, they share no elements:is empty, that is, they share no elements:

AAB = B =

The The differencedifference between two sets A and B between two sets A and B contains exactly those elements of A that are contains exactly those elements of A that are not in B:not in B:

A-B = {x | xA-B = {x | xA A x xB}B}Example: Example: A = {a, b}, B = {b, c, d}, A-B = {a}A = {a, b}, B = {b, c, d}, A-B = {a}

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Set OperationsSet Operations

The The complementcomplement of a set A contains exactly of a set A contains exactly those elements under consideration that are those elements under consideration that are not in A: not in A: AAcc = U-A = U-A

Example: Example: U = U = NN, B = {250, 251, 252, …}, B = {250, 251, 252, …} BBcc = {0, 1, 2, …, 248, 249} = {0, 1, 2, …, 248, 249}

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Set OperationsSet OperationsTable 1 in Section 1.5 shows many useful Table 1 in Section 1.5 shows many useful

equations.equations. How can we prove AHow can we prove A(B(BC) = (AC) = (AB)B)(A(AC)?C)?

Method I:Method I: xxAA(B(BC)C) xxA A x x(B(BC)C) xxA A (x (xB B x xC)C) (x(xA A x xB) B) (x (xA A x xC)C)

(distributive law for logical expressions)(distributive law for logical expressions) xx(A(AB) B) x x(A(AC)C) xx(A(AB)B)(A(AC)C)

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Set OperationsSet OperationsMethod II: Method II: Membership tableMembership table

1 means “x is an element of this set”1 means “x is an element of this set”0 means “x is not an element of this set” 0 means “x is not an element of this set”

11111111111 1 11 1 1

11111111001 1 01 1 0

11111111001 0 11 0 1

11111111001 0 01 0 0

11111111110 1 10 1 1

00001100000 1 00 1 0

00110000000 0 10 0 1

00000000000 0 00 0 0

(A(AB) B) (A(AC)C)AACCAABBAA(B(BC)C)BBCCA B CA B C

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Set OperationsSet Operations

Every logical expression can be transformed Every logical expression can be transformed into an equivalent expression in set theory and into an equivalent expression in set theory and vice versa.vice versa.