SETS

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

Information: SETS