Himpunan (Inggris'an)

7/29/2019 Himpunan (Inggris'an)

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1) Finite and infinite set:

A set is finite if you can list all its elements and infinite otherwise.

2)Ways to define a set:

a. With a verbal description: All sets above are described verbally when we say, " The set of all

bla bla bla "

b. A listing of all members separated by commas with braces ({ and }):

A listing of set 1 is written as: {a, b,c,d,e,f,....,z}

c. Set-builder notation:

Set 1 and set 4 can be written as { x / x is a letter of the modern English alphabet} and { x / x is a

type of sausage}

{ x / x is a letter of the modern English alphabet} is read, " The set of all x such that x is a letter

in the modern English alphabet

We use capital letters such as A, B, and so forth to denote sets

For example, you could let A be the set of all positive numbers less than 10.

We use the symbol to indicate that an object belongs to a set and the symbol to indicate that

an object does not belong to a set

For example, if A is the set of all positive numbers less than 10, then 2 A, but 12 A

3)Empty set orNull set : A set that has no element and is denoted by { } or

For example, {x / x is a human being who have lived 10,000 years} is an empty set because it is

impossible to find at least one human being who have lived so long

4)Two sets are equal if they have exactly the same element

For example, { x / x is a number between bigger than 1 and less than 5} and { 2, 3, 4} are equal

5)

Subset of a set

Set B is a subset of a set A if and only if every object of B is also an object of A.

We write B A

By definition, the empty set( { } or ) is a subset of every set


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B = { a, b, c}

A = { a, b, c, f}

U = { a, b, c, f}

Since all elements of B belong to A, B is a subset of A

6) Universal set:

The set that contains all elements being discussed

In our example, U, made with a big rectangle, is the universal setSet A is not a proper subset of U because all elements of U are in subset A

Notice that B can still be a subset of A even if the circle used to represent set B was not inside

the circle used to represent A. This is illustrated below:


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7) Union of Sets

Definition:

Given two sets A and B, the union is the set that contains elements or objects that belong to

either A or to B or to both

We write A B

Basically, we find A B by putting all the elements of A and B together

A = { x / x is a number bigger than 4 and smaller than 8}B = { x / x is a positive number smaller than 7}

A = { 5, 6, 7} and B = { 1, 2, 3, 4, 5, 6}

A B = { 1, 2, 3, 4, 5, 6, 7}

notice that 5 and 6 were written only once although it would be perfectly ok to write them twice

The graph below shows the shaded region for the union of three sets


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8) Intersection of Sets

Definition:

Given two sets A and B, the intersection is the set that contains elements or objects that belong toA and to B at the same time

We write A B

Basically, we find A B by looking for all the elements A and B have in common

Example :

A = { x / x is a number bigger than 4 and smaller than 8}B = { x / x is a positive number smaller than 7}

A = { 5, 6, 7} and B = { 1, 2, 3, 4, 5, 6}

A B = {5, 6}

The graph below shows the shaded region for the intersection of two sets


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The graph below shows the shaded region for the intersection of three sets

9) Complement of Sets

Definition:

Given a set A, the complement of A is the set of all element in the universal set U, but not in A.

We can write Ac

example :

Let B = {1 orange, 1 pinapple, 1 banana, 1 apple}

Let U = {1 orange, 1 apricot, 1 pinapple, 1 banana, 1 mango, 1 apple, 1 kiwifruit }

Again, we show in bold all elements in U, but not in B

Bc = {1 apricot, 1 mango, 1 kiwifruit}

The graph below shows the shaded region for the complement of set A


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This ends the lesson about the complement of a set.

10) Difference of Sets

Definition:

Given set A and set B the set difference of set B from set A is the set of all element in A, but not

in B.

We can write A B

Example:A = { x / x is a number bigger than 6 and smaller than 10}

B = { x / x is a positive number smaller than 15}

A = {7, 8, 9} and B = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}

Everything you see in bold above are in B only.

B A = {1, 2, 3, 4, 5, 6, 10, 11, 12, 13, 14}

The graph below shows the shaded region for A B and B A


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