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Revision Notes on SET THEORY We use {π₯1 , π₯2 , π₯3 , … , π₯π } to denote a set, where π₯1 , π₯2 , π₯3 , … , π₯π are the elements of a set. Let us call this set, π΄. Hence, using the language of mathematics we write, π₯1 ∈ π΄ , π₯2 ∈ π΄, π₯3 ∈ π΄, … , π₯π ∈ π΄ where ∈ is read as ‘belongs to’ or ‘is an element of’. If there are ‘no’ elements in the set then we simply write { } which represents an ‘empty set’. The elements are distinct objects (anything that we may want to study). Let us suppose that Q is the set of chapters from the ‘textbook’ by Alpha C. Chiang that will appear in Mid 1, then Q = {Ch. 2,Ch. 3}. Here, the universal set π is the set of all chapters in the ‘textbook’. Hence, π = {Ch. 1, Ch. 2, Ch. 3, ..., Ch. 20}. Where, ‘…’ means ‘so on and so forth in the obvious fashion’. The sets that we have written so far are examples of writing a set using the technique of enumeration (simply listing the elements of the set). Another technique of writing a set is by description. E.g. Q = { x | x is a chapter that will appear in the Mid 1} and π = {π₯ |π₯ ππ π πβπππ‘ππ ππ π‘βπ ′π‘ππ₯π‘ππππ ′ ππ¦ ππ. πΆβππππ} Properties of SETS: 1) Order of the elements in a set is irrelevant (does not matter). Hence, {a,b} = {b,a} 2) Elements of a set cannot appear multiple times. Hence {a,a,b} is NOT a set. A set is a subset of another set (the superset) of ALL the elements in the set ∈ another set. Hence, Q is a subset of U or Q ⊂ U * or U ⊃ Q **. *Read: Q is a subset of U ** U is a superset of Q. The empty set is a subset of ALL sets. And if two sets are equal then they are subsets of each other. So far we have discussed some relations between sets. Now let us consider some operations that we may perform given two sets. We can combine two sets and this called the ‘union’ (π π¦ππππ: ∪) of two sets. For example, if πΌ = {π₯|π₯ ππ ππ πππ‘ππππ} = {…,-3,-2,-1,0,1,2,3,…} i.e. πΌ is the set of all integers (all positive and negative whole numbers and zero), and πΉ = {π₯|π₯ ππ π πππππ‘πππ} i.e. πΉ is the set of ALL fractions. Then, πΌ ∪ πΉ gives us the set of ALL rational numbers (those numbers that can be represented as the ratio of two integers i.e. ALL integers and fractions). And combining (unionising) the set of ALL rational and irrational numbers gives us the set of ALL real numbers. The key point to note here is that the ‘new set’ we obtain from the union of two sets will contain only those elements that belong to either of the given two sets. Using a simpler example, given, π΄ = {1,2,3} and π΅ = {3,4,5} then, π΄ ∪ π΅ = {1,2,3,4,5}. We can also take the intersection (∩) of the two sets which gives us a new set containing only those elements that are common to the given sets. Using our last example, π΄ ∩ π΅ = {3}. The complement of a ‘given set’ is a new set that contains only those elements that are not in the ‘given set’ but are in the universal set , e.g. complement of the set (Q) that contained all the chapters included in Mid 1 gives us a new set π ′ = πΜ = {πβ. 1, πβ. 4, πβ. 5, … , πβ. 20} = {π₯|π₯ ππ π πβπππ‘ππ ππ ′π‘ππ₯π‘ππππ ′ π‘βππ‘ π€πππ πππ‘ ππππππ ππ πππ 1} = {π₯|π₯ ∈ π πππ π₯ ∉ π}. The last operation on sets, we shall consider here, is that of the multiplication of two sets. When we multiply two sets we get a new set containing ordered pairs (π₯, π¦). In this case the ordered pairs are the elements of the sets. The first variable ′π₯′ of the pair MUST be from the first set and the second variable ′π¦′ MUST be from the second set. E.g. given, π΄ = {1,2} and π΅ = {3,4,5} then, π΄ × π΅ = {(1,3), (1,4), (1,5), (2,3), (2,4), (2,5)}. We have already seen an application of SET THEORY where we used it to derive the set of all real numbers. If we represent the set of all real numbers on a straight horizontal line then we get the real line (x axis) and if we represent take another vertical real line (y axis) which intersects the horizontal real line at ‘0’ then we get the famous rectangular (Cartesian) coordinate plane, which is used to perform graphical analysis of the relationship between two variables in almost every field of study imaginable… Some BASIC CONCEPTS related to the rectangular coordinate plane: If you see a line or curve (drawn in the rectangular plane) that is upward sloping (red line/supply curve in the figure below) then this means the variables are directly related (as one variable increases so does the other). In the figure below, as the quantity increases (as you move right along the horizontal axis), the price increases as well…vice versa. A downward sloping curve or line indicates that the variables are inversely related (as one increases the other decreases). Hence the demand curve in the figure below is downward sloping, since price and quantity demanded is inversely related…as one increases the other decreases. Source: www.faculty.washington.edu/danby/bls324/surplus.html The rate of increase or decrease is given by the gradient of the line or curve where ππππππππ‘ =β³ π¦/β³ π₯ . For example, in the figure above when quantity supplied changes from 0 to 5 units the price increases 5 from $0 to $ 5. Hence, Δπ₯ = 5 − 0 = 5 πππ Δπ¦ = 5 − 0 = 5, gradient= 5 = 1. When the quantity demanded changes from 0 to 5 units then the price drops from 10 to 5 units. Here, Δπ¦1 = 5 − 10 = 5 −5. Therefore, gradient = − 5 = −1. 1 Change = Final value – Initial value