SETS
A set is any collection or group of objects. The data that contains all objects is called the universal set (U). Each object is called a member or element () of the set. A subset () is a collection that is part of the universal set. The empty set is set that contains no objects and represented by the symbol .
Two ways of defining a set:
- Roster method
listing the elements
eg. Even number less than 10. Answer: {2, 4, 6, 8}
- Set-builder notation
describe the elements
eg. {a, e, i, o, u} Answer: {x|x are vowels in the alphabet}
Well-defined and Not Well-defined set:
eg. {teachers in grade 7} Answer: Yes, because it is clear that a teacher is teaching in grade 7.
{a popular basketball player} Answer: No, because some people may consider a player popular while others not.
Cardinality of each set:
- Number of elements in a set deals in answering the question "how many?" is a cardinal number, written n(A).
- eg. A={grade 7 subjects} Answer: n(A) = 12
Finite or Infinite set:
- Finite set if the set is empty or if it can be placed into a one-to-one correspondence
number of elements is whole number
eg. The set of weekdays
- Infinite set if the set is not countable
eg. The set of whole numbers
Equal or equivalent sets:
- Equal set if and only if they contain EXACTLY the same elements
eg. A = {apple, mango} and Z = {mango, apple}
- Equivalent set if and only if there is a one-to-one correspondence between the sets
eg. B = {red, yellow, blue} and Y = {blue, white, black}
Subset
- if and only if every element of A is also an element of B
- written: A B
- eg. Let U = {Cyclops, Wolverine, Storm, Beast}
A = {Storm, Beast} ->
B = {Proffesor X, Cyclops, Wolverine} ->
Proper subset
- if and only if every element of A is also in B and B contains at least one element that is not in A
- written: A B
- eg. Let U = {Circle, Triangle, Rectangle, Square}
A = {Triangle, Rectangle} ->
B = {Square, Triangle, Circle, Rectangle} ->
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