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Sets, , Introduction, Chapter.Snapshot, , ip ow mathemencs! language. evervihmg m this universe whether living or non-living is, called ar abiect. we consider 2 collectiin of objects given in such a way that it is possible, to tell bevond down. whether a grven obrect is m the collection under consideration or not, + Introduction, ther suct ¢ collecnon of abvects ix called a well-defined collection of objects. — + Set, Set + Notations”, + Representation of Sets, , A set w ¢ well-defined collecnon of objects. By ‘well-defined collection of objects’, it + Types of Sots, , mean: that we car definitely decide whether a given particular object belongs to a given, : + Venn Diagram, , colleches or not. The objects. clements and members of a set are synonymous terms., , ¢ Operations on Sets, © Example 1. The set of intelligent students in a class ts + Laws of Aigebra of Sets, tac uli set (hy) a well-defined collection F j Séiw Practical, . iis + Formulae to Solve Prac!, fe; 0 fntee set {dj not a well-defined collection Problems.on'Union'and, Intersection of Sets, , Bal. 6: Smee me crtenor to aetermning tne mialigence Of a stucent May vary from person to, penser 20 Tres @ 10! 6 web-Gefrec comeChOn Of ODjECts, © Example 2. Which of the following is the collection of first five prime numbers?, (hy (2,35, 7,104, 19 13.5.7. 1b 134 {dj {1, 2; 3, 4, 5), , Sal. &, 6.6 cee mat wet ive pre numbers are (2.3.6.7 11}, , Sets or asually donated by cupstal iemers 4, 8, etc., and theu elements by small, , fotters a, b, <, etc., , Let Abe any set of oipjocts and let a be 3 member of A, then we write a € A and read it, , as ‘a belongs to A” of “6 is an clement uf A’ or 'u ip member of A” fa is not an object, of Athen we write a € 4 and read ws “a Sure pot beluag Wu 4” oF “4 is mol an element of, , A’ Of '@is not a member of 4’., , Representation of Sets, , A s¢tis often represented in ane of the following two forus ., /— @ Roster form or Tabular form (it) Set builder form :
Page 3 : Equal Sets, , Two sets A and B are said to be equal, if every, clement of A in also an clement of B and every element, Of B is alvo an element of A. Thun, fx Ao xe Band, , Y& B=» ye A, then A and B are equal sets and we can, write A= B,, , @ Example Ll. The set A= (x:x6 N and, 1<x <7) és equal to, , (a) {2,3, 4,5, 6} (b) 41, 2,3, 4, 5. 6}, () (2, 3,4, 5,6, 7} (d) 41, 2, 3,4, 5,6, 7), , Sol. (0) Whenever, we have to show that two sols A and B, are equal, show that A Band Bc A, then A» B, , Equivalent Sets, , Two finite sets A and B are said to be equivalent, if, n(A) = n(B)., , Clearly, equal sets are equivalent, but equivalent, Sets need not to be equal,, , Equivalence of two sets is denoted by the symbol, ‘~". Thus, if A and B are equivalent sets, we write, A ~ B which is read as ‘A ig equivalent to 2’,, , @ Example 12. The sci {1, 2, 3,4, 5} is, equivalent to, , (a) {x:xeall vowels} (b) (2,3, 4}, () (1,2,4,5) d) {1,2,3}, Soll. (a), , Subset and Superset, , The set B is said to be subset of set A, if every, element of set B is also an element of set A., , Symbolically, we write it as, BC A or A > B, where A, is superset of B., , (i) B ¢ Ais read as B is contained in A or B is subset, of A or A is superset of B., (ii) AD B is read as A contains B or B isa subset of A., Evidently, if A and B are two sets such that Ac B, => xe A, then B is subset of A. The symbol ‘=’ stands, for ‘implies’. We read it as ‘x belongs to B” implies that, ‘x belongs to A’., , © Example 13. The set A= {1,2,3} is the subset, of, (a) {1,2,4, 5}, () (1, 2, 3, 7, 8}, Sol. (c), , Proper Subset, , The set B is said to be a proper subset of set A, if, every element of set B is an clement of 4, whereas, every element of A is not an element of B,, , (6) {1, {2,3}, 4, 5}, (4) {{1, 25,3, 4}, , We write it as Bc A and read it a6 B is @ proper, subset of A, Thus, 2 is a proper subset of A, if every, element of # is an clement of A and there is atleast one 3, element in A which is not in B., , Observe that Ac A i.e. every set is a subset of, itvelf, but not a proper subset., , @ Example 14. The set {1,2} is the proper, , subset of, , (a) (24 (hy 11, 3,45, () N,2, 35 (d) (41, 2}, 3,4}, Sol. (c), , Intervals as Subsets of R, , , , Leta, be R anda <b, then, (i) An open interval denoted by (a, b) is the set of all, , real numbers such that (a, b) = {x€ R:a<x <b}, , (ii) A closed interval denoted by [a, 5] is the set of all, real numbers such that (a, b]=txe R:a Sx Sb}, , (iii) Interval closed at one end and open at the other, are given by[a, b)={xeRasx<by, and (a, b]={xER:a<x<b}, , @ Example 15. The subset of R as intervals, te:x€R,-12<x<-10} is, (a) (-12, -10] (b) [-12, -10], (©) (-12, -10) (a) [-12, -10), Sol. (c) The given subset is helong to open interval., {x: xe R, -12.< x< 10} is (12, - 10},, , © Example 16. Let R be set of points inside a, rectangle of sides a and b (a,b >1) wth two sides, , along the positive direction of X-axis and Y-axis,, then, , (a) R={(x, y):0Sx<sa,0s y<b}, (b) R=((x, y):0Sx<a,0< y<b}, () R={(x, y):0Sx<a,0< y<b}, (d) R={(x, y):0<x<a,0< y<b}, Sol. () Since, (x, y)both lies in first quadrant., , “. %y> Oand less than side a and b because R lies, inside the rectangle., , R={(x y):0<x<a,0< y<b}, Power Set, , The set formed by all the subsets of a given set A is, called the power set of A. It is usually denoted by P(A)., @ Example 17. 1 set 4 = {,2,3}, then P(A) is, , equal to, , (a) {, {1}, {2}, (3}, {1,2}, (2, 31 (1 3, , () M1}, {2}, (3), {1,2}, 42, 3}, 43,, , (C) {. {1}, {2}, {3}, {1,2}, €2, 3}, {3,1 1,2, 3}, , (@) M{1}, {2}, 43}, (1, 2, 3}}, , Sol. (0)
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Some Results on Subsets, (i) Every set is a subset of itself., ii empty set is a subset of every set. ;, a 7 total number of subset of finite set containing, nelements is 2"., Comparable Sets, , sets Aand B are said to be comparable, if one, snmis asubset of the other i.e. either 4 cB or, , 3, , of, Bch, e gsample 18. The ser {1, 2, 3} is comparable, wi 2 ASI @) X11, 2}, 4,3}, yh 32 (4) {1,2,3, 43, sol. {0 Since. {1, 2. 3} is the subset of {1 2,3, 4}, Set, , In any discussion of set theory, there is always a set, that contains all the sets under consideration i.e. it is a, , et of each of the given sets. Such a set is called, the universal set and is denoted by U., , @ Example 19. Which of | the following set is the, sniversal set of sets A= {2. 4,5}, B = {13,5}?, C=6.5.7.11}. D= 24810}, , fa) {1,2.3, 5,6, 7,8, 9, 10}, (1 11.2, 3.4, 5,6, 7, 8, 10}, (e) (3.2, 3,4, 5, 7, 8, 10, 11}, di 1, 2,3. 4,6, 7,8, 10,11}, Sol. (c), , Venn Diagram, Venn diagrams are the diagrams which represent, the relationship between sets., , © Example 20. The set of natural numbers is a, subset of whole numbers which is a subset of, , , , , , , , , , , , , , , , , , imegers. Then, correct representation of it through, Venn diagram is, U, fa) 7) zZ, (e) (4) None of these, fol, (a), , , , , , Operations on Sets, , Now, we introduce some Operations, construct néw sets from the given ones,, , i,, , ON Sets to, , “Union of sets. The union of two sew aa, B, denoted by A U B is the set of all those, elements, each one of which is either in 4 o, in B or in both A and B., , , , , , , , , , , , | Thus, AUB =fx:xe Aorxe B}, , | Clearly x€AUB, , > xed, , or xeB, , and X€@AUB |, , => xé€A and xr¢B, , In the figure, the shaded part represents, A U B, It is evident that ACAUB,BCAUB., , @ Example 21. [f two sets A = {l,2,3} and, B= {1,3,5, 7}, then AUB is equal to, , (a) {1,2,3, 7} ) (1, 2,3, 5, 7, 8}, (¢) {1, 2, 3,5, 7} (d) {1, 2, 3,4, 5, 6, 7}, Sol. (c), , , , , , , , id. | Intersection of sets The intersection of two |, sets A and B, denoted by 4 > B is the set of all, , elements, common to both 4 and B., , , , U, , |, |, , , , A 8, , , , , , Thus,, Clearly,, =>, , ANB={x:xe AandxeB}, , xEANB |, , xed and xeB \, andx€ ANB = xed or xE8, , In the figure, the shaded part represents 4 B.|, , Wis evident that AnBG4ANBSE. |, , @ Example 22. /f two sets A= (123.4) and, B= (2,45) then A7B is equal 0, (a) {1,3} (b) (2,44, () 3,4} (d) {t, 6}, , Sol. (0) Since, 2 and 4 are the common elements in sets A, and
