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Puzzle

A census taker interviews a woman in a house.

Census Taker: Who lives here?Woman: My husband and I and our three daughters.CT: What are the ages of your your daughters?W: The product of their ages is 36 and the sum of their ages is the

house number.

The census taker looks at the house number, thinks, and says

CT: You havent given me enough information to determine theages.W: Oh, youre right. Let me also say that my eldest daughter isasleep upstairs.CT: Ah! Thank you very much!

What are the ages of the daughters?

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Assumption: the ages are nonnegative integers.

Fact: the product of ages is 36.

LetPbe the set of all ordered triples (x, y, z) with x y zwhose product is 36.

Note that we allow the numbers to be equal because the womanmay have twins or triplets.

Assumption: if not twins/triplets, age must differ by at least 1.

So, P={(36, 1, 1), (18, 2, 1), (12, 3, 1), (9, 4, 1), (9, 2, 2), (6, 6, 1), (6, 3, 2), (4, 3, 3)

If house number is 21, what are the ages of the daughters?

What does it mean that CT did not have enough information?

How does the womans answer resolve the problem?

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Sets

A set is a collection of distinct objects.

The objects in a set are itselements or members.

Whenxis an element ofA, we writex A and say xbelongs toA.

Whenxis not in A, we writex6 A.

Examples:

A={1, 5, 9, 21}

B= {True, False}

C= {Red,Blue, 10,Yes,Penguin}

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Sets

If every element ofA belongs to B, then A is a subset ofB, andBcontains A; we write A B orB A.

SetsA and Bare equal, writtenA = B, if they have the sameelements.

The empty set, written, is the unique set with no elements.A proper subset of a set A is a subset ofA that is not A itself.

The power set of a setA is the set of all subsets ofA.

Example:

LetA = {0, 1}

The power set ofA is {, {0}, {1}, {0, 1}}.

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Some Important Sets

1. N ={1, 2, 3, 4, . . .}, the set of natural numbers

2. Z ={. . . ,2,1, 0, 1, 2, . . .}, the set of integers

3. Q ={p/q : p, q Z, q6= 0}, the set of rational numbers

4. R, the set ofreal numbers

These sets satisfy the following containment relationship:N Z Q R.

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Sets

Set builder notation

This is used to define or describe sets that are too big or complexto list between braces.

This appears in the following form:

X= {expression : rule}.

So, X is the set of all things of the form expression that satisfythe rule.

Examples:

Q ={p/q : p, q Z, q6= 0}, the set of rational numbers Even numbers: {2k : k Z} Odd numbers: {2k+ 1 : k Z} [n] = {1, 2, . . . , n}= {k N : k n} The interval(6, 1) = {x R : 6< x< 1} A={x R : x2 + 5x< 6}

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Sets

An ordered pairis a list (x, y) of two things x and y.

The Cartesian product of two setsA and Bis another set,

denotedA Band defined as A B= {(a, b) : a A, b B}.

A k-tuple is a list of the form (x1, x2, . . . , xk).

Ak ={(x1, x2, . . . , xk) : xi A for eachi= 1, 2, . . . ,k}

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Sets

Set Operations

LetA and Bbe two sets. Then,

Union A B= {x : x A or x B}

Intersection A B= {x : x A and x B}

Difference A B= {x A : x6 B}

Two sets aredisjoint if their intersection is the empty set.

If the setA is contained in someuniverseU, then thecomplementofA, writtenAc, is the set of elements inUthat are not in A.

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Properties/Laws of Sets

LetA,B,Cbe sets and letWbe a universal set containing allthree sets.

Commutativity: A B= B A and A B= B A

Associativity: A (B C) = (A B) C andA (B C) = (A B) C

Distributivity: A (B C) = (A B) (A C) andA (B C) = (A B) (A C)

A =A, A W= A

A = , A W =W

A Ac = , A Ac =W

(A B)c =Ac Bc

(A B)c =Ac Bc

IfA Ba nd B C, thenA C

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Set Inclusion

The following are equivalent (meaning, if one of the statements istrue, then they are all true):

A B

A B= A

A B= B

A B= Bc Ac

Note that the word and refers to the intersection and the wordor refers to the union of sets.

LetEbe the set of even numbers, letObe the set of oddnumbers, and letPbe the set of prime numbers.

Example: Ifxis even and prime, then x= 2 as E P= {2}.

Example: Ifxis even or odd, thenxcan be any integer

write a counter example to prove that the given statement is wrong .

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Set Inclusion

Example:

LetA = {x R : x2 x}, letB= {0, 1}, and C= [0, 1].

i. Show that B A but that A 6 B.

ii. Show that C A.

Answer.

(i) To prove thatB A, we need to check that all members ofBare members ofA. As Bonly has two members, we can checkindividually.

As 02 = 0 0 A and 12 = 1 1 A, it follows that B A.

To show that A 6 B, it is enough to find just one member ofAwhich is not a member ofB. Indeed, as 1/2 Abut 1/2 6 B, wehave A 6 B.

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Set Inclusion

As C has infinitely many elements, we cannot individually checkthat all its members also belong toA.

We need a way to check everything simultaneously.

An argument of this type is usually of the form:

pick an arbitrary elementx C(do not specify which one, it

could be any) use a property that all items in C have

show that this impliesx A

Answer.

(ii). Pick an arbitrary item x C. Note that this means0 x 1. As x 0, we can multiply both sides of the inequalityx 1 by x, and we get x2 x. Therefore,x satisfies the conditionfor membership inA (that is, x A). It follows that C A.

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Equality of Sets

It is possible to give different descriptions to describe the same set.

For example, letA={countries that share a border with Nepal} andB= {countries with a population of more than 1 billion people}.

One can check thatA =B= {China, India}.

There are many instances when we want to determine whether twosetsA and B, that are defined in different ways, are actually equalor not.

How can we determine this?

1. A= Bif and only ifA B and B A.

2. To show A B, we must show that everyx A also belongsto B.

3. IfA 6 B, enough to find one thing in A which is not in B.

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Equality of Sets

Exercises:

Show that{x R :x2 + 5x< 6}= (6, 1)

De Morgans Law: (A B

)

c

=Ac Bc

LetA = {(x, y) : xy

+ yx 2}and B= {(x, y) : xy> 0}. Show

that A = B.

LetS= {x : |x/(x+ 1)| 1} and let T= [1/2,). Showthat S= T.

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Exercise

LetWbe the set of ordered pairs of nonzero numbers. That is,W ={(x, y) : x, y6= 0}.

LetA = {(x, y) W : xy

+ yx 2} and

B= {(x, y) W : xy> 0}.

Show that A =B.

We have to show thatA Ban d B A.

We will showBc Ac and B A.