Intersection Of Sets (1966)

your film is now ready to be shown this is the Central High baseball team in the language of mathematics they are said to constitute a set let us call this set T each player is called a member of the set and is said to belong to the set a set is a collection of things that have something in common which we can describe these members all have in common the fact that they play baseball for Central High School this player is a member of set T because he plays on the Central High baseball team we can see that the manager does not belong to setti because he does not play on the team now consider just the pitchers they have in common the fact that they pitch for the team therefore they also constitute a set which we will call set P since each member of set p is also a member of set T we say that P is a subset of T this can be represented by the use of this symbol which means subset of in general if we select members from a given set they form a new set which is a subset of the given set let's form some more subsets of set T the two player who are catchers and the four players who are outfielders also form sets C and O since all members of both teams belong to set T they are both subsets of T similarly we can indicate the set of infielders this boy plays first base this one second base and this one third base this one plays short stop and this one is a utility infielder these five all have in common the fact that they play in the infield so we may refer to them as a set which we will designate by the letter i in the set I or infield the players five 6 7 and 8 constitute the regular infield which we can call set r then R is a subset of I a method has been devised to show pictorially the relationships between different sets and subsets to show the relationship of sets r i and T we represent each as a circle and arrange them to form a figure called a van diagram we know that every member of R is also a member of I I and this is represented by this Vin diagram with circle R entirely inside of circle I since every member of I is a member of t or the entire team it can be seen that every member of R is also a member of T in other words if R is a subset of I and I is a subset of T then r R is also a subset of T there are many other things that the members of our baseball team may have in common here the players marked with L are left-handed batters likewise the players marked with r are right-handed batters we can now form the set of all left-handed batters and the set of all right-handed batters you may have already noticed a different relationship between these two sets that players four and 10 belong to both sets since they have this fact in common they therefore constitute a third set which is called the intersection of sets L and R in general the intersection of two sets is a third set each of whose members belongs to each of the two given sets at the same time if we simplify the ven diagram by omitting the numerals then Circle L represents the first set the left handers and Circle R represents a second set the right-handers the region common to both circles represents the intersection of the two sets this third set we will call set Z using the symbol for intersection the intersection of sets L and R is set z z then is the set of players who can bat either left-handed or right-handed there are many more possible intersections in the sets we have formed for example we may consider the set of left-handed batters together with a set of pitches players numbered one and four are common to both sets and form a third set which is the intersection of sets L and P if we call this third set y we may say that the intersection of sets L and P is y and their intersections are found almost anywhere consider the spectators watching the game those sitting in the odd vertical rows form a set which we will call set o consider also the set of Spectators in the even horizontal rows we will call this set e those marked with both o and E constitute a third set if we call this set D we may say that the intersection of O and E is D in another situation what is the intersection of set s consisting of Spectators in the first vertical row and set F whose members are the spectators in the fourth vertical rle since the two sets have no members in common we say that the intersection of sets S and F is the null set or empty set we indicate the null set with this symbol let's look now at all the possible patterns of intersections for two sets again we can demonstrate this by using ven diagrams in this first situation the intersection of sets A and B may be the empty set since neither set has members common to the other set and here the intersection of A and B may be part of both A and B if we mark this set with letter C then C is the intersection of A and B and all its members are common to both A and B also C is a subset of both A and B in a third situation the intersection of sets A and B can be the set B itself and all its members belong to set a b then is also a subset of a and finally the intersection of the two sets can be the same as both sets in this case where A and B are the same set their intersection is a or b all members of set a are the same as all members of set B also a is a subset of itself and in general every set is a subset of itself now let's illustrate the four patterns of intersections with specific examples in the first pattern the intersection of two sets is the empty set the intersection of the set of outfielders set o and the set of catchers set C is the empty set the intersection of the set of numerals in set o and the set of numerals in set C is also the empty set this can also be shown by using a number line the intersection of the set of these points and the set of these points on the number line is the empty set the intersection of the set of Spectators in the horizontal row and the set of Spectators in another horizontal row is the empty set the intersection of the set of points on two parallel lines is the empty set or in the language of geometry you would say that parallel lines never meet in the second pattern where the intersection of two sets is a subset of each of them we can illustrate this by showing the intersection of the set of infielders set I and the set of left-handers set L as the set consisting of the one player that belongs to both I and L the left-handed infielder the intersection of the set of numerals in set I and the set of numerals in set l is the set consisting of the one numeral common to both sets the intersection of the set of these points on the number line and the set of these points is the set consisting of a single point the intersection of the set of Spectators in a horizontal Row in the bleachers and the set of Spectators in a vertical row is the set consisting of the one person common to both rows the intersection of the set of points on line a and the set of points on line B is the set consisting of this single point this may also be stated in Geometry by saying that two lines that are not parallel determine a point in the third pattern the intersection of sets A and B is set B for example the intersection of the set of catchers and the set of all players in the team is the set of catchers which is a subset of the entire team the intersection of the set of just the numerals from 1 to 15 and the set of numerals 10 and 11 is the set 10 and 11 which is a subset of the larger set the intersection of the set of these points on the number line and the set of these points is the set consisting of points 10 and 11 finally in the fourth pattern the intersection of two sets can be the same set thus the intersection of A and B is a itself or B since A and B are the same set the intersection of a set of outfielders with a set of outfielders with numerals on their uniforms is the set of outfielders itself the intersection of the set of these points on the number line with the set of these points is the set of the same points let us review some of the concepts of intersection of sets a set is a collection of things that have something in common which we can describe for in instance these players have in common the fact that they play baseball for the same team if we select members from a given set they form a new set which is a subset of the given set for instance each of these infielders is also a member of the entire team there are four possible patterns of intersections for two sets first the intersection of sets A and B may be the empty set second the intersection of sets A and B may be a subset which is part of both A and B third the intersection of sets A and B may be set B then B is also a subset of a and fourth if a and b are the same their intersection will also be the same set we have studied the different patterns of intersection for only two sets but it is also possible for three sets to intersect or four or any number with each additional set the patterns of intersection become more and more complicated so that the number of possible intersections of sets is unlimited