ACTIVITY 1 AIM : To find the number of subsets of a given set and verify that if a set has n number of elements, then the total number of subsets is 2”. Material Required Paper, different coloured pencils. Prerequisite Knowledge 1. Concept of set and its types. 2. Concept of subset. Method of Construction 1, Take the empty set (say) X which has no element. 2, Take a set (say) X; which has one element (say) x, 3. Take a set (say) Xp which has two elements (say) x; and x, 4, Take a set (say) X, which has three elements (say) x, and x5, Demonstration 1. Represent X; as in Fig. (1.1). Here the possible subsets of X, is X, itself only, represented symbolically by 6. The number of subsets of Xp is 1 = 2”. Figure 4 2. Represent X, as in Fig, (1.2). Here the possible subsets of X, are ¢, {x}. The number of subsets of X, is 2=2 if Ni Figure 12 | Laboratory Manual Mathematies Class 11thHere the possible subsets of A, are 6, {2 4, Represent X, as in Fig, (1.4). Here, the subsets of X, are & {a,x} and {x,, 2), x}. The number of subsets of X, is 8= 2°. 5, Continuing in this way, the number of subsets of set X containing n elements x, 2, Observation 1, The number of subsets of Xy is Fas ie es 2. The number of subsets of X, is ...... 3, The number of subsets of X, is, 4, The number of subsets of X; is 5, The number of subsets of Xiq is 6. The number of subsets of X,, is Result We have learnt how to find the number of subsets of a set and verify that if a set has n number: elements, then the total number of subsets is 2” Application We can use these activity for finding the number of subsets of a given set. Mea Jug) (x1, xe} The number of subsets of X; is 4= (xy). fagh (23h Ot toh aaah is|AIM : To represent set theoretic operations using Venn Diagrams. Material Required Cardboard, white thick sheet of paper, different coloured pencil, scissors, gum. Prerequisite Knowledge 1. Universal set. 2, Concept of Venn diagrams. 8. Concept of operations on sets. Method of Construction 1. Firstly, cut rectangular strips from a sheet of paper and paste them on a cardboard. Write the gymbol U in the left/right top corner of each rectangle. 2, After that, draw two circles A and Beach inside of the rectangular strips and shade/colour @ifferent portions as shown in Fig. (8.1) to Fig. (10). Demonstration 1. U denotes the universal set represented by the rectangle. 2. Circles Aand Brepresent the subsets of the universal set U as shown in the Fig. (3.1) to Fig. (3.10). 3, A’and B’ denote the complement of the set Aand set B respectively as shown in the Fig. (3.3) and Fig. (3.4). 4. In the Fig. (3.1), shaded portion represents A UB (4) Figures Laboratory Manual Mathematics Class 11th5. In the Fig. (3.2), shaded portion represents AM B Figure 32 6. In the Fig. (3.8), shaded portion represents A’. x Figure 3.3 7. In the Fig. (3.4), shaded portion represents BY Figure 3.4 8. In the Fig. (3.5), shaded portion represents (A. By’. U 3 Figure 3.5 9. In the Fig. (3.6), shaded portion represents (A U BY = Figure 3.6 10, In the Fig, (3.7), shaded portion represents 4’ 9 Bwhich is similar B— A U Figure 37 Laboratory Manual Mathematics Class Ath11, In the Fig, (8.8), shaded portion represents A’ U B. 12. Fig. (8.9) shows An B= 4 18, Fig. (3.10) shows Ac B Figure 3.10 Observation 1, Coloured portion in Fig. (8.1) represents .. Coloured portion in Fig. (3.2) represents . Coloured portion in Fig. (3.3) represents Coloured portion in Fig. (3.4) represents .. Coloured portion in Fig. (3.5) represents Coloured portion in Fig. (3.6) represents .. Coloured portion in Fig. (3.7) represents Coloured portion in Fig. (3.8) represents .. Fig. (3.9) shows that (A 9 B) = 10. Fig. (8.10) represent A... Sernaae B. Result We have learnt how to represent set theoretic operations by using Venn diagrams. Application Set theoretic representation of Venn diagrams are used in Logic and Mathematics. Laboratory Manual Mathematics Class 11thAIM :To verify distributive law for three given non-empty sets A, Band C, that is, AU(BOC) =(AU B)a(AUC) Material Required Cardboard, white thick sheets of paper, different coloured pencils, scissors, gum. Prerequisite Knowledge 1. Concept of universal set. 2. Concept of union of sets. 8. Concept of intersection of sets. 4, Concept of complement of sets. 5. Distributive law. Method of Construction 1. Firstly, cut five rectangular strips from a white sheet of paper and paste them on the cardboard in such a way that three of the rectangles are in horizontal line and two of the remaining rectangles are also placed horizontally in a line just below the above three rectangles. Write the symbol U in the left/right top corner of each rectangle as shown in Fig. (4.1) to Fig. (4.5). 2. Drawn three circles and mark them as A, Band Cin each of the five rectangles as shown in the below figures, 3, Colour/shade the portions as shown in the figures. Laboratory Manual Mathematics Class 11thDemonstration 1, Hore, U denotes the universal set represented by the rectangle in each figure, 2, Subsets of the universal set U are represented by circles A, Band C. {— u a Aus Figure 42 AvEnAL Figure 4 Figure 5 Aven) | shaded portions in Fig. 4.2 represents AU B 3. In Fig. 4.1 shaded portion represents B 0 C, s! ‘AU (Bo C)and shaded portion in Fig. 4.5 Fig. 4.3 represents Au C, Fig. 44 represents represents (4 U B) (AU ©). Observation 1. Coloured portion in Fig. (4.1) represents 2. Coloured portion in Fig. (4.2) represents 3. Coloured portion in Fig. (4.3) represents 4, Coloured portion in Fig, (4.4) represents 5. Coloured portion in Fig. (4.5) represents «1.1. 6. The common shaded portions in Fig. (4.4) and Fig. (4.5) are TAU (BO O)= ses From the coloured portion in Fig, (4.4) and Fig. (4.8) are the same Hence, the distributive law is verified. Result We have learnt distributive for three given non-empty sets and how to verify it Application Distributivity property of set operations is used in the simplification of problems involving set operations. Note In the same way, the other distributive law A 0 (BU C)= (Aq B)U (An C) can also be verified, Gs) Laboratory Manual Mathematics Class 11thACTIVITY 5 AIM : To identify a relation and a function. Material Required Cardboard, battery, electric bulbs of two diffe switches, erent colours, testing screws, tester, electrical wires a Prerequisite Knowledge 1. Concept of ordered pair. 2. Concept of relation. 3. Concept of function. Methods of Construction 1. Firstly, take a piece of cardboard of suitable size and paste a white paper on it. 2. Drill eight holes on the left side of board in a column and mark them as A, B,C, D, E,F,G and: as shown in the Fig. (5.1). see an 3, Drill seven holes on the right side of the board in a column and mark them as P,Q, R, S,7,U. and V as shown in the Fig (5.1). After that, fix bulbs of same colour in the holes A, B,C, D, B, F,@ and H. . Similarly, fix bulbs of the other colour in the holes P, Q, R, 8, 7,U and V. In the bottom of the board fix testing screws marked as 1, 2, 3, ... 8 Complete the electrical circuits in such a manner that a pair of corresponding bulbs, one from | each column glow simultaneously. rage Laboratory Manual Mathematics Class 11thNR MeN tn ements). ety 8. These pairs of bulbs will give ordered pairs, which will constitute a relation which in turn may/may not be a function [see Fig. (6.1)]. 9, Arelation f from a non-empty set Ato a non-empty set Bis said to be a function, if every element of set A has one and only one image in set B Demonstration 1. Bulbs at A, B,..., H, along the left column represent domain and bulbs along the right column at P,Q, R, -«., V represent co-domain. 2, We can obtain different order pairs out of given eight screws by using two or more testing screws. In Fig. (6.1), all the eight screws have been used to give different ordered pairs such as (AP), (B, RB), (C, (A, B), (EQ), ete. . By choosing different ordered pairs make different sets of ordered pairs. Observation 1, In Fig. (6.1), ordered pairs are . 2. These ordered pairs constitute a .. 3, The ordered pairs (A, P), (B, R), (C, @) (E, @, (D, 1), (G, 7), (F, U), (HU) constitute a relation which is also a... 4, The ordered pairs (B, R), (C, Q), (D, 7), (E, 8), (E, Q) constitute a which is not a Result We have learnt how to identify a relation and a function. Application The activity can be used to explain the concept of a relation and a function. It can also be used to explain the concept of one-one, onto functions. Necessary Concepts Related to Activity 1. Ordered Pairs Ifa pair of elements written in circular bracket and grouped together in a particular order, then such a pair, is called an ordered pair. Ordered pair is written by listing two objects in the specific order, separated by comma (:) and enclosing the pair in parentheses. eg. The ordered pair of two elements a and bis denoted by (a, b); a is first element (or first component) and bis second element (or second component). ‘Two ordered pairs are equal if their corresponding elements are equal. ie (a,b)=(¢,d) > a=cand b=d 2, Relation Let A and Bbe two non-empty sets. Then, a relation R from set A to set Bis a subset of cartesian product Ax Bie. RCAXB. ‘The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in Ax B. The second element is called image of first element. If(a, be R, then we write it as ab and we say that a is related to 6 by the relation R. If(a,b)¢R, then we write it as ab and we say that a is not related to 6 by the relation R. citraet ert Pee OER Torr G@