OBJECTIVE MATHEMATICS FOR ENGINEERING ENTRANCE EXAMINATIONS
SETS, RELATIONS AND FUNCTIONS (II) SET-BUILDER FORM IN THIS METHOD, WE WRITE THE SET BY SOME SPECIAL PROPERTY OR PROPERTIES SATISFIED BY ALL ELEMENTS AND WRITE IT AS: A {X: X HAS THE PROPERTY P(X)} EXAMPLES: TABULAR FORM SET-BUILDER FORM 1 A {K, R, I, S, H} 2 B {1, 4, 9, 81} 3 C {1, 1 , , , 4 9 } A {X: X IS A LETTER OF THE WORD ‘KRISH’} B {X: X N2, NN AND N D 10} C {X: X N AND N N} NOTES: (I) THE ORDER IN WHICH THE ELEMENTS OF TWO SETS ARE WRITTEN IS IMMATERIAL THUS, A {2, 3, 5} {2, 5, 3} {3, 2, 5} (II) THE REPETITION ELEMENTS IN A SET HAS NO MEANING THUS, B {2, 3, 5} {2, 2, 3, 5, 5} {2, 2, 3, 3, 5, 5, 5} 3 TYPES OF SETS (I) NULL SET OR VOID SET OR EMPTY SET A SET CONTAINING NO ELEMENT IS CALLED THE EMPTY SET OR THE NULL SET OR THE VOID SET IT IS DENOTED BY I OR { } EXAMPLES: (I) A {X: 1 < X < 2 AND X N} (II) B {X: X IS EVEN PRIME NUMBER GREATER THAN 2} (III) C {X: X2 47 AND X IS AN INTEGER} (IV) D {X: X IS THE 27TH LETTER OF ENGLISH ALPHABET} (V) E {X: X2 4 AND 3X 7} NOTES: (I) NULL SET IS UNIQUE (II) A SET CONTAINING AT LEAST ONE ELEMENT IS CALLED A NON-EMPTY SET (II) SINGLETON SET A SET CONTAINING ONLY ONE ELEMENT IS CALLED A SINGLETON SET EXAMPLES: (I) A {X | X < 2 AND X N} {1} (II) B {X | X2 25 0 AND X > 0} {5} (III) C {X | X IS AN EVEN PRIME NUMBER} {2} (IV) D {X | X IS THE CAPITAL OF INDIA} {DELHI} (III) PAIR SET A SET CONTAINING TWO ELEMENTS IS CALLED A PAIR SET EXAMPLES: (I) A {X: X < 3 AND X N} {1, 2} (II) B {X: X2 10X + 21 0} {3, 7} (III) C {X: X IS A LETTER OF THE WORD TAT} {T, A} (IV) A {X:X2 121} X2 121 Ÿ X R 11 ? A {11, 11} IS A PAIR SET (IV) FINITE SET AND INFINITE SET A SET WHICH IS EMPTY OR CONSISTS OF A DEFINITE NUMBER OF ELEMENTS IS CALLED FINITE OTHERWISE, THE SET IS CALLED INFINITE EXAMPLES (FINITE SETS) (I) A {1, 4, 9, 16, 25, 36} > @ N (A) ' (II) B {X: X2 100 0 AND X Z} > @ N (B) ' (III) C {X: X IS A LETTER OF ENGLISH ALPHABET} > @ N (C) ' (IV) D {X: X IS A DAY OF THE WEEK} > @ N (D) ' EXAMPLES (INFINITE SETS): (I) THE SET OF ODD NATURAL NUMBERS (II) THE SET OF ALL POINTS ON A LINE SEGMENT (III) THE SET OF ALL STRAIGHT LINES (DRAWN IN A PLANE) PASSING THROUGH A GIVEN POINT SETS, RELATIONS AND FUNCTIONS REMARKS (I) WE REPRESENT INFINITE SET IN THE ROSTER FORM BY WRITING A FEW ELEMENTS WHICH INDICATE STRUCTURE OF THE SET FOLLOWED (PRECEDED) BY THREE DOTS FOR EXAMPLE, N {1, 2, 3, } IS THE SET OF NATURAL NUMBERS AND Z {, 3, 2, 1, 0, 1, 2, 3, } IS THE SET OF INTEGERS IN ROSTER FORM (II) THE INFINITE SET WHICH DO NOT FOLLOW ANY PARTICULAR PATTERN CANNOT BE REPRESENTED IN ROSTER FORM BECAUSE ITS ELEMENTS DO NOT FOLLOW ANY PARTICULAR PATTERN CARDINAL NUMBER (OR ORDER) OF A FINITE SET THE NUMBER OF DIFFERENT ELEMENTS IN A FINITE SET A IS CALLED THE CARDINAL NUMBER (OR ORDER) OF A IT IS DENOTED BY N(A) EXAMPLES: (I) LET A {1, 3, 5, 7, 9}, THEN N(A) 5 (II) LET B {X: (X 1) (X + 2) (X 3) 0}, THEN N (B) 3 (III) LET C {X: X IS A LETTER OF THE WORD KOLKATA} CLEARLY, C {K, O, L, A, T} SO N(C) 5 (V) EQUAL SET TWO SETS A AND B ARE SAID TO BE EQUAL, IF THEY HAVE EXACTLY THE SAME ELEMENTS WE WRITE A B NOTE: TWO EQUAL SETS ARE ALWAYS EQUIVALENT BUT NOT THE CONVERSE EXAMPLES: (I) P {X: X IS A LETTER OF THE WORD ‘BHART’} Q {X: X IS A LETTER OF THE WORD ‘BHRAT’} SOL P {B, H, A, R, T}, Q {B, H, R, A, T} THUS, P Q (II) SHOW THAT P Q, IF P {X: X2 1 0 AND X N}, Q {X: X SIN LT O T T T } SOL P {1}, Q {1} THUS, P Q (VI) EQUIVALENT SETS TWO FINITE SETS A AND B ARE SAID TO BE EQUIVALENT IF THEY HAVE THE SAME CARDINAL NUMBERS, I E , A AND B ARE EQUIVALENT œ N (A) N (B) EXAMPLES: LET A {A, E, I, O, U} AND B {1, 2, 3, 4, 5} SINCE N (A) N (B) 5, THEREFORE A AND B ARE EQUIVALENT NOTES: (I) THE SYMBOL ‘Ÿ’ IS READ AS ‘IMPLIES’ THUS, X Ÿ Y IS READ AS X IMPLIES Y (II) THE SYMBOL ‘ œ ’ IS READ AS ‘IMPLIES AND IMPLIED BY’, œ IS ALSO READ AS ‘IF AND ONLY IF’ (BRIEFLY WRITTEN AS ‘IFF’) (VII) SUBSETS LET A, B BE ANY TWO SETS, THEN THE SET A IS SAID TO BE A SUBSET OF SET B IF EVERY ELEMENT OF A IS ALSO AN ELEMENT OF B, AS WE WRITE, A Ž B OR B ‹ A (READ AS ‘A IS CONTAINED IN B’ OR ‘B CONTAINS A’) B, IS CALLED SUPER SET OF A THUS, A Ž B IF X A Ÿ X B EXAMPLE: LET A {1, 2, 3, 4}, B {3, 4, 9} AND C {1, 2, 3, 4, 5, 6} THEN A Ž C [ ' EVERY ELEMENT OF A IS CONTAINED IN C] AND B Œ C ['9 B BUT 9 C, I E ALL THE ELEMENTS OF B ARE NOT CONTAINED IN C] (VIII) SUPER SET IF A Ž B, THEN B IS CALLED SUPER SET OF A AND WE WRITE B ‹ A (IX) PROPER SUBSET LET A BE ANY SET AND B BE A NON-EMPTY SET, THEN A IS CALLED A PROPER SUBSET OF B IF EVERY ELEMENT OF A IS ALSO AN ELEMENT OF B AND THERE EXIST AT LEAST ONE ELEMENT IN B WHICH IS NOT AN ELEMENT OF A THUS, IF A Ž B AND A Z B, THEN A IS CALLED PROPER SUBSET OF B, AND WE WRITE A B SETS, RELATIONS AND FUNCTIONS EXAMPLE: N W Z Q R REMARKS: (I) THE EMPTY SET IS A SUBSET OF EVERY SET (II) EVERY SET IS A SUBSET OF ITSELF (III) THE EMPTY SET AND THE SET ITSELF ARE CALLED IMPROPER SUBSETS (IV) FOR ANY TWO SETS A AND B, A = B œ A Ž B AND B Ž A (V) TWO SETS A AND B ARE SAID TO BE COMPARABLE IF ONE OF THEM IS A SUBSET OF THE OTHER, I E , EITHER A Ž B OR B Ž A INTERVALS AS SUBSETS OF R: THE SET OF REAL NUMBERS LET A, B R AND A < B, THEN, WE CAN DEFINE: (I) OPEN INTERVAL (A, B) {X R: A < X < B} (II) CLOSED INTERVAL [A, B] {X R: A D X D B} (III) RIGHT HALF OPEN INTERVAL (OR SEMI-CLOSED SEMI-OPEN INTERVAL) [A, B) {X R: A D X < B} (IV) LEFT HALF OPEN INTERVAL (OR SEMI-OPEN SEMI-CLOSED INTERVAL) (A, B} {X R:A < X D B} ON REAL NUMBER LINE, VARIOUS TYPES OF INTERVALS DESCRIBED AS SUBSET OF R ARE SHOWN BELOW: (A, B) [A, B] [A, B) (A, B] A B A B A B A B REMARKS: (I) THE NUMBER (B A) IS CALLED THE LENGTH OF ANY OF THE INTERVALS (A, B), [A, B], [A, B), OR (A, B] (II) ALL INTERVALS HAVE INFINITE MEMBERS (X) POWER SET THE COLLECTION (OR SET) OF ALL SUBSETS OF A SET A IS CALLED THE POWER SET OF A IT IS DENOTED BY P(A) OR 2A FOR EXAMPLE, IF A {1, 2, 3}, THEN P(A) {I}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3} CLEARLY, IF A HAS N ELEMENTS, THEN ITS POWER SET CONTAINS EXACTLY 2N ELEMENTS (XI) UNIVERSAL SET A SET WHICH IS SUCH THAT ALL THE SETS UNDER CONSIDERATION ARE ITS SUBSET, IS CALLED THE UNIVERSAL SET IT IS DENOTED BY U OR S OR ] OR : REMARKS: (I) UNIVERSAL SET IS A KIND OF ‘PARENT SET’ AND EVERY SET IN A PARTICULAR CONTEXT, IS A SUBSET OF THE UNIVERSAL SET (II) THE CHOICE OF UNIVERSAL SET IS NOT UNIQUE, I E , UNIVERSAL SET MAY VARY FROM SITUATION TO ANOTHER (XII) VENN DIAGRAM MOST OF THE PROPERTIES OF SETS CAN BE UNDERSTOOD AND VERIFIED WITH THE HELP OF DIAGRAMS KNOWN AS VENN DIAGRAMS NAMED AFTER AN ENGLISH LOGICIAN JOHN VENN (1834-1883) IN THESE DIAGRAMS, THE UNIVERSAL SET (U) IS REPRESENTED BY A RECTANGULAR REGION AND ITS SUBSETS BY CIRCLES OR ELLIPSES INSIDE THE RECTANGLE WE REPRESENT INTERSECTING SETS BY INTERSECTING CIRCLES AND DISJOINT SETS BY DISJOINT CIRCLES IN VENN DIAGRAMS, THE ELEMENTS OF THE SETS ARE WRITTEN IN THEIR RESPECT CIRCLES OR ELLIPSES EXAMPLES: CONSIDER THE UNIVERSAL SET U {1, 2, 3, 4 5, 6, 7, 8, 9} AND THE SETS A {1, 2, 3, 4, 5} AND B {3, 4, 5, 8} THE GIVEN SETS ARE REPRESENTED BY MEANS OF VENN DIAGRAM IN THE FIGURE A B SETS, RELATIONS AND FUNCTIONS 4 OPERATION OF SETS (I) UNION OF TWO SETS THE UNION OF TWO SETS A AND B IS THE SET CONSISTING OF ALL THE ELEMENTS WHICH ARE EITHER IN A OR IN B OR IN BOTH THUS, A ‰ B {X: X A OR X B} CLEARLY, X A ‰ B Ÿ X A OR X B AND X A ‰ B Ÿ X A AND X B THE UNION OF A AND B IS DENOTED BY A ‰ B AND READ “A UNION B” EXAMPLES: (I) IF A {A, B, C, D} AND B {C, D, E, F}, THEN A ‰ B {A, B, C, D, E, F} (II) LET A {1, 2, 3, 4, 5, 6} AND B {2, 4, 6, 8} THEN A ‰ B {1, 2, 3, 4, 5, 6, 8} NOTES: (I) X A ‰ B œ X A OR X B (II) X A ‰ B œ X A AND X B (III) IF A Ž B, THEN A ‰ B B (IV) THE UNION OF SEVERAL SETS VIZ A1 ‰ A2, ‰, … ‰ AN IS SOMETIMES ABBREVIATED AS N I I A * (II) INTERSECTION OF TWO SETS THE INTERSECTION OF TWO SETS A AND B, WRITTEN AS A ˆ B (READ AS ‘A INTERSECTION B’) IS THE SET CONSISTING OF ALL THE COMMON ELEMENTS OF A AND B THUS, A ˆ B { X: X A AND X B} CLEARLY, X A ˆ B Ÿ X A AND X B, AND X A ˆ B Ÿ X A AND X B EXAMPLE: IF A {A, B, C, D} AND B {C, D, E, F}, THEN A ˆ B {C, D} NOTES: (I) X A ˆ B œ X A AND X B (II) X A ˆ B œ X, X A OR X B (III) IF A Ž B, THEN A ˆ B A (IV) TWO SETS A AND B ARE SAID TO BE DISJOINT SETS, IF THEY HAVE NO COMMON ELEMENTS, I E , A ˆ B I 5 DISJOINT SETS TWO SETS A AND B ARE SAID TO BE DISJOINT, IF A ˆ B I, I E , A AND B HAVE NO COMMON ELEMENT EXAMPLE: IF A {1, 2, 5} AND B {3, 4, 6}, THEN A ˆ B I, SO A AND B ARE DISJOINT SET 6 DIFFERENCE OF TWO SETS IF A AND B ARE TWO SETS, THEN THEIR DIFFERENCE A B IS DEFINED AS: A B {X: X A AND X B} SIMILARLY, B A {X: X B AND X A} EXAMPLE: IF A {1, 2, 3, 4, 5} AND B {1, 3, 5, 7, 9}, THEN A B {2, 4} AND B A {7, 9} NOTES: (I) A B IS A SUBSET OF A; AND B A IS SUBSET OF B (II) A B Z B A (I E , THE DIFFERENCE OF A AND B IS NOT IDENTICAL WITH THE DIFFERENCE OF B AND A) (III) A B A B ˆ C (I E , THE DIFFERENCE OF A AND B IS IDENTICAL WITH THE INTERSECTION OF A AND B COMPLEMENT) IMPORTANT RESULTS (A) A B Z B A (B) THE SETS A B, B A AND A ˆ B ARE DISJOINT SETS (C) A B Ž A AND B A Ž B (D) A I A AND A A I 7 SYMMETRIC DIFFERENCE OF TWO SETS THE SYMMETRIC DIFFERENCE OF TWO SETS A AND B, DENOTED BY A ' B, IS DEFINED AS SETS, RELATIONS AND FUNCTIONS A ' B (A B) ‰ (B A) {X: X A OR X B BUT X A ˆ B} EXAMPLE 1: IF A {1, 2, 3, 4, 5} AND B {1,3, 5, 7, 9} THEN A ' B (A B) ‰ (B A) {2, 4} ‰ {7, 9} {2, 4, 7, 9} EXAMPLE 2: LET A {2, 4, 6, 8, 10}, B {8, 10, 11, 12} HERE A B {2, 4, 6} AND B A {11, 12} ? A ' B (A B) ‰ (B A) {2, 4, 6, 11, 12} 8 COMPLEMENT OF A SET THE COMPLEMENT OF A SET A IS THE SET OF ALL ELEMENTS OF THE UNIVERSAL SET S, WHICH DO NOT BELONG TO A THIS IS WRITTEN AS AC OR A OR AC AND READ “A PRIME” OR “A BAR” OR “A COMPLEMENT” RESPECTIVELY ILLUSTRATION: LET THE UNIVERSAL SET BE S {1, 2, 3, 4, 5, 6} AND A {2, 4} THEN AC {1, 3, 5, 6} NOTE: (I) A A S ‰ C AND A A ˆ I C (I E , THE UNION OF A SET AND ITS COMPLEMENT IS THE UNIVERSAL SET, AND ITS INTERSECTION OF A SET AND ITS COMPLEMENT IS THE NULL SET) (II) S AND S I I C C (I E , UNIVERSAL SET AND NULL SET ARE COMPLEMENTARY TO EACH OTHER) (III) (A C ) C = A (I E , THE COMPLEMENT OF THE COMPLEMENT OF A SET IS THE SET ITSELF) 9 ALGEBRA OF SETS (I) IDEMPOTENT LAWS FOR ANY SET A, WE HAVE (A) A ‰ A A (B) A ˆ A A (II) IDENTITY LAWS FOR ANY SET A, WE HAVE (A) A ‰ I A (B) A ˆ I I (C) A ‰ U U (D) A ˆ U A (III) COMMUTATIVE LAWS FOR ANY TWO SETS A AND B WE HAVE (A) A ‰ B B ‰ A (B) A ˆ B B ˆ A (IV) ASSOCIATIVE LAWS FOR ANY THREE SETS A, B AND C, WE HAVE (A) A ‰ (B ‰ C) (A ‰ B) ‰ C (B) A ˆ (B ˆ C) (A ˆ B) ˆ C (V) DISTRIBUTIVE LAWS FOR ANY THREE SETS A, B AND C, WE HAVE (A) A ‰ (B ‰ C) (A ‰ B) ˆ (A ‰ C) (B) A ˆ (B ˆ C) (A ˆ B) ‰ (A ˆ C) (VI) FOR ANY TWO SETS A AND B, WE HAVE (A) P (A) ˆ P (B) P (A ˆ B) (B) P (A) ‰ P (B) Ž P (A ‰ B), WHERE P (A) IS THE POWER SET OF A (VII) IF A IS ANY SET, WE HAVE A A C C (VIII) DEMORGAN’S LAWS (A) (A ‰ B)C = AC ˆ BC (B) (A ˆ B)C = AC ‰ BC (C) A – (B ‰ C) = (A – B) ˆ (A – C) (D) A – (B ˆ C) = (A – B) ‰ (A – C) 10 IMPORTANT RESULTS ON OPERATIONS ON SETS (I) A Ž A ‰ B, B Ž A ‰ B, A ˆ B Ž A, A ˆ B Ž B (II) A B A ˆ BC (III) (A B) ‰ B A ‰ B (IV) (A B) ˆ B I SETS, RELATIONS AND FUNCTIONS (V) A B B A C C (VI) A Ž B œ BC Ž AC (VII) (A ‰ B) ˆ (A ‰ B) = A (VIII)A ‰ B (A B) ‰ (B A ) ‰ (A ˆ B) (IX) A (A B) A ˆ B (X) A B B A œ A B (XI) A ‰ B A ˆ B œ A B (XII) A ˆ ( B ' C) ( A ˆ B) '(A ˆ C) 11 SOME BASIC RESULTS ABOUT CARDINAL NUMBER IF A, B AND C ARE FINITE SETS AND U BE THE UNIVERSAL SET, THEN (I) N(AC) = N(U) – N(A) (II) N(A ‰ B) N(A) + N(B) N( A ˆ B) (III) N(A ‰ B) N(A) + N(B), WHERE A AND B ARE DISJOINT NON-EMPTY SETS (IV) N(A B ) N(A) N(A B) ˆ ˆ C (V) N(A B ) N(A B) N(U) N(A B) C ˆ ‰ C C ‰ (VI) N(A B ) N(A B) N(U) N(A B) C ‰ ˆ C C ˆ (VII) N(A B) N(A) N(A B) ˆ (VIII) N(A B) ˆ N(A B) N(A B ) N(A B) ‰ ˆ ˆ C C (IX) N(A B C) N(A) N(B) N(C) ‰ ‰ N(A B) N(B C) N(C A) ˆ ˆ ˆ N(A B C) ˆ ˆ (X) IF A1, A2, A3, … AN ARE DISJOINT SETS, THEN N(A1 ‰ A2 ‰ A3 ‰‰ AN) N(A1) + N(A2) + N(A3) + … N(AN) (XI) N(A ' B) NUMBER OF ELEMENTS WHICH BELONG TO EXACTLY ONE OF A OR B 12 CARTESIAN PRODUCT OF TWO SETS LET A AND B BE TWO NON-EMPTY SETS THEN THE CARTESIAN PRODUCT A U B IS THE SET OF ALL ORDERED PAIRS OF ELEMENTS FROM A AND B, I E , A U B {(A, B): A A, B B} NOTES: (I) IF A I OR B I, THEN WE DEFINE A U B I (II) A U B Z B U A (III) IF A HAS N ELEMENTS AND B HAS M ELEMENTS, THEN A U B HAS M N ELEMENTS (IV) IF A1, A2, , AU ARE U NO-EMPTY SETS, THEN THEIR CARTESIAN PRODUCT, IS DEFINED AS ^ ` I 1, I I I 1A A A ,,A ;A FOR ALL I U U S $ 13 SOME USEFUL RESULTS ON CARTESIAN PRODUCT IF A, B, C ARE THREE SETS, THEN (I) A U (B ‰ C) (A U B) ‰ (A U C) (II) A U (B ˆ C) (A U B) ˆ (A U C) (III) A U (B C) (A U B) (A U C) (IV) (A U B) ˆ (S U T) (A ˆ S) U (B ˆ T) WHERE S AND T ARE TWO SETS (V) IF A Ž B, THEN (A U C) Ž (B U C) (VI) IF A Ž B, THEN (A U B) ˆ (B U A) A2 (VII) IF A Ž B AND C Ž D THEN A U C Ž B U D (VIII)IF A Ž B, THEN A U A Ž (A U B) ˆ (B U A) (IX) IF A AND B ARE TWO NON-EMPTY SETS HAVING N ELEMENTS IN COMMON, THEN A U B AND B U A HAVE N2 ELEMENTS IN COMMON (X) A U B B U A IF AND ONLY IF A B (XI) A B C (A B) A C C U ‰ U ˆ U C C (XII) A B C (A B) A C C U ˆ U ‰ U C C PROBLEM SET 1 THE NUMBER OF NON-EMPTY SUBSETS OF THE SET {1, 2, 3, 4} IS (A) (B) (C) (D) [CET (KARNATAKA), 1997)] SOL NUMBER OF NON-EMPTY SUBSETS OF {1, 2, 3, 4} 15 ANS (A) SETS, RELATIONS AND FUNCTIONS 2 IF A AND B ARE ANY TWO SETS, THEN A ‰ (A ˆ B) IS EQUAL TO (A) BC (B) AC (C) B (D) A [CET (KARNATAKA), 1996)] SOL A ‰ (A ˆ B) A ['A ˆB Ž A] ANS (D) 3 SET A AND B HAVE 3 AND 6 ELEMENTS RESPECTIVELY WHAT CAN BE THE MINIMUM NUMBER OF ELEMENTS IN A ‰ B" (A) [CET (KARNATAKA), 1996)] SOL N(A ‰ B) N(A) + N(B) N(A ˆ B) 3 + 6 N(A ˆ B) SINCE MAXIMUM NUMBER OF ELEMENTS IN A ˆ B 3, ? MINIMUM NUMBER OF ELEMENTS IN A ‰ B 9 3 ANS (B) 4 A {X: X Z X} REPRESENTS (A) {X} (B) {1} (C) { } (D) {0} [CEET (HARYANA), 1998)] SOL CLEARLY, A I { } ANS (C) 5 IF P, Q AND R ARE SUBSETS OF A SET A, THEN R U (PC ‰ QC)C EQUALS (A) (R U P) ˆ (R U Q) (B) (R U Q) ˆ (R U P) (C) (R U P) ‰ (R U Q) (D) NONE OF THESE SOL R U (PC ‰ QC) C R U [(PC) C ˆ (QC) C] R U (P ˆ Q) (R U P) ˆ (R U Q) ANS (A) 6 IN A CLASS 100 STUDENTS, 55 STUDENTS HAVE PASSED IN MATHEMATICS AND 67 STUDENTS HAVE PASSED IN PHYSICS, THEN THE NUMBER OF STUDENTS WHO HAVE PASSED IN PHYSICS ONLY IS (A) [CEE (DELHI), 1993] SOL N(M) 55, N(P) 67, N(M ‰ P) NOW, N(M ‰ P) N(M) + N(P) N(M ˆ P) Ÿ 100 55 + 67 N(M ˆ P) N(M ˆ P) 122 100 ANS (A) 7 GIVEN THE SETS A {1, 2, 3}, B {3, 4}, C {4, 5, 6}, THEN A ‰ (B ˆ C) IS (A) {1, 2, 3, 4, 5, 6} (B) {1, 2, 4, 5} (C) {1, 2, 3, 4} (D) {3} SOL B ˆ C {4} ? A ‰ (B ˆ C) {1, 2, 3, 4} ANS (C) 8 LET A {A, B, C}, B = {B, C, D}, C {A, B, D, E}, THEN A ˆ (B‰ C) IS (A) {C} (B) {A, B, C} (C) {B, C, D} (D) {A, B, D, E} [CEET (HARYANA), 1997)] SOL B ‰ C {A, B, C, D, E} ? A ˆ (B ‰ C) {A, B, C} ANS (B) 9 THE SET A {X: X R, X2 16, AND 2X 6} EQUALS (A) I (B) {14, 3, 4} (C) {3} (D) {4) [CET (KARNATAKA), 1995] SOL WE HAVE, X2 X + ALSO, 2X Ÿ X THERE IS NO VALUE OF X WHICH SATISFIES BOTH THE ABOVE EQUATIONS THUS, THE SET A CONTAINS NO ELEMENTS ? A I ANS (A) SETS, RELATIONS AND FUNCTIONS 10 IF THE SET A HAS P ELEMENTS, B HAS Q ELEMENTS, THEN THE NUMBER OF ELEMENTS IN A U B IS (A) P + Q + (D) P + Q [CET (KARNATAKA), 1996] SOL AS A HAS P ELEMENTS AND B HAS Q ELEMENTS, A U B HAS PQ ELEMENTS ANS (B) 11 IF X AND Y ARE TWO SETS SUCH THAT N(X) 17, N (Y) 23, N(X ‰ Y) 38, THEN N(X ˆY) IS (A) (D) NONE OF THESE SOL WE HAVE N(X) 17, N(Y) 23, N(X ‰ Y) 38 N(X ‰ Y) N(X) + N(Y) N(X ˆ Y) Ÿ 38 17 + 23 N(X ˆ Y) Ÿ N(X ˆ Y) = 40 38 ANS (A) 12 X AND Y ARE TWO SETS SUCH THAT X HAS ELEMENTS, X ‰ Y HAS 60 ELEMENTS AND X ˆ Y HAS 10 ELEMENTS, THEN NUMBER OF ELEMENTS Y HAS (A) (D) NONE OF THESE SOL HERE, N(X) 40, N(X ‰ Y) 60, N( X ˆ Y) N(X ‰ Y) N(X) + N(Y) N(X ˆ Y) Ÿ 60 40 + N(Y) Ÿ N(Y) 60 + 10 40 ANS (C) 13 IF X AND Y ARE TWO SETS SUCH THAT X HAS ELEMENTS, Y HAS 32 ELEMENTS, AND X ˆ Y HAS 11 ELEMENTS, THEN NUMBER OF ELEMENTS X ‰ Y HAS (A) (D) NONE OF THESE SOL WE HAVE N(X) 21, N(Y) N(X ˆ Y) N(X ‰ Y) N(X) + N(Y) N( X ˆ Y) N(X ‰ Y) 21 + 32 42 HENCE X ‰ Y HAS 42 ELEMENTS ANS (A) 14 IF N(U) 700, N(A) 200, N(B) 300 AND N(A ˆ B) 100, THEN N(A B ) ˆ C C IS EQUAL TO (A) (D) NONE OF THESE SOL N(A B ) (A B)C ˆ ‰ C C N(U) N(A B) ‰ N(A B) ‰ NOW, N(A ‰ B) N(A) + N(B) N(A ˆ B) = 200 + 300 400 HENCE, N(A B ) ˆ C C 700 300 ANS (B) 15 IF N(U) 60, N(A) 35, N(B) 24 AND N(A B)C ‰ 10, THEN N(A ˆ B) IS (A) (D) NONE OF THESE SOL WE HAVE N(A ‰ B) N(U) N(A B)C ‰ 60 50 NOW, N(A ‰ B) N(A) + N(B) N(A ˆ B) 35 + 24 9 ANS (C) 16 IN A GROUP OF 65 PEOPLE, 40 LIKE CRICKET, LIKE BOTH CRICKET AND TENNIS THE NUMBER OF PERSON LIKING TENNIS ONLY AND NOT CRICKET IS (A) (D) NONE OF THESE SETS, RELATIONS AND FUNCTIONS SOL U LET A BE THE SET OF PEOPLE WHO LIKE CRICKET AND B THE SET OF PEOPLE WHO LIKE TENNIS THEN N(A ‰ B) N(A) N(A ˆ B) ? N(A ‰ B) N(A) + N(B) N(A ˆ B) 65 40 + N(B) Ÿ N(B) 65 40 + NUMBER OF PEOPLE WHO LIKE ONLY TENNIS N(B) N(A ˆ B) 35 ? NUMBER OF PEOPLE WHO LIKE ONLY TENNIS ANS (C) 17 IN A GROUP OF 1000 PEOPLE, THERE ARE PEOPLE WHO CAN SPEAK HINDI AND 400 PEOPLE WHO CAN SPEAK ENGLISH THEN NUMBER OF PERSONS WHO CAN SPEAK HINDI ONLY IS (A) (D) NONE OF THESE SOL U H E HERE, N(H ‰ E) N(H) N(E) N(H ‰ E) N(H) + N(E) N(H ˆ E) 1000 750 + 400 N(H ˆ E) Ÿ N(H ˆ E) 1150 1000 NUMBER OF PEOPLE WHO CAN SPEAK HINDI ONLY N(H E )C ˆ N(H) N(H ˆ E) 750 150 ANS (C) 18 IF X AND Y ARE TWO SETS AND XC DENOTES THE COMPLEMENT OF X, THEN (X) ˆ (X ‰ Y) C EQUALS (A) X (B) Y (C) I (D) NONE OF THESE SOL X X Y X X Y ˆ ‰ C ˆ C ˆ C [ BY DEMORGAN’S LAW ' (A B) (A B )] ‰ C C ˆ C (X X ) Y ˆ C ˆ C Y I ˆ C = I ANS (C) 19 LET A {X: X IS A MULTIPLE OF 3} AND B {X: X IS A MULTIPLE OF 5} THEN A ˆ B IS GIVEN BY (A) {3, 6, 9, } (B) {5, 10, 15, 20, } (C) {15, 30, 45 } (D) NONE OF THESE SOL SINCE X A ˆ B œ X A AND X B œ X IS A MULTIPLE OF 3 AND X IS A MULTIPLE OF 5 œ X IS A MULTIPLE OF 5 HENCE A ˆ B {X | X IS A MULTIPLE OF 15} {15, 30, 45,} ANS (C) 20 IF X {8N 7N 1 N N} AND Y {49(N 1)/N N}, THEN (A) X Y (B) Y X (C) X Y (D) NONE OF THESE SOL WE HAVE, 8N 7N (7 + 1)N 7N SETS, RELATIONS AND FUNCTIONS N N ˜ ˜ ˜ ! N N N C ˜ N N N N C ˜ ˜ ˜ N N N N C C ˜ ˜ FOR N T FOR N 1, 8N 7N 1 8 7 1 THUS, 8N 7N 1 IS A MULTIPLE OF 49 FOR N T 2, AND 0 FOR N 1 HENCE, X CONSISTS OF ALL POSITIVE INTEGRAL MULTIPLE OF 49 OF THE FORM 49 KN WHERE N N N N N N K C C ˜ ˜ TOGETHER WITH ZERO ALSO Y CONSISTS OF ALL POSITIVE INTEGRAL MULTIPLES OF 49 INCLUDING ZERO THEREFORE, X Y ANS (A) 21 IF A {1, 3, 5, 7, 9, 11, 13, 15, 17}, B {2, 4, 6, ,18} AND N IS THE UNIVERSAL SET, THEN A A B B C ‰ ‰ ˆ C IS (A) A (B) B (C) N (D) NONE OF THESE SOL WE HAVE, A B B A ‰ ˆ C A B B A A A N ‰ ˆ C ‰ C ‰ C ANS (C) 22 SUPPOSE A1, A2,… A30 ARE THIRTY SETS, EACH WITH FIVE ELEMENTS AND B1, B2,, BN ARE N SETS EACH WITH THREE ELEMENTS LET N I J I U A U B S IF EACH ELEMENT OF S BELONGS TO EXACTLY TEN OF THE AJCS AND EXACTLY NINE OF THE BJCS THEN N (A) (D) NONE OF THESE SOL GIVEN A’S ARE THIRTY SETS WITH FIVE ELEMENTS EACH SO, I I N A U ¦ (1) IF THE M DISTINCT ELEMENTS IN S AND EACH ELEMENT OF S BELONGS TO EXACTLY 10 OF THE AICS, WE HAVE N A 10M ¦ (2) FROM (1) AND (2), WE GET 10 M OR M 15 SIMILARLY I I J N B 3 N AND N B 9 M ¦ ¦ ? 3N = 9 M 9M N 3M Ÿ U HENCE, N ANS (C) 23 IF A {(X, Y): X2 + Y2 25} AND B {(X, Y): X2 +9Y2 144}, THEN N (A ˆ B) IS (A) (C) NONE OF THESE SOL X2 + Y2 25 IS A CIRCLE WITH RADIUS 5 UNIT X2 + 9Y2 X Y (12) IS AN ELLIPSE OF WHICH THE MAJOR AXIS IS ALONG THE X-AXIS THE LENGTH OF THE MAJOR AXIS IS 24 UNIT AND IT IS GREATER THAN THE DIAMETER OF THE CIRCLE THE MINOR AXIS IS ALONG THE Y-AXIS AND ITS LENGTH 8 UNIT THE MINOR AXIS IS LESS THAN THE DIAMETER OF THE CIRCLE SO, THE ELLIPSE INTERSECTS THE CIRCLE AT 4(FOUR) POINTS I E , N (A ˆ B) 4 ANS (D) XC YC Y X SETS, RELATIONS AND FUNCTIONS 24 IF A {P, Q, R, S}, THEN THE NUMBER OF PROPER SUBSETS OF A U A IS (A) 216 (B) 216 SOL HERE, N(A U A) 16 SO N(P(A U A)) 216; ONE OF THEM IS A U A ITSELF, WHICH IS AN IMPROPER SUBSET OF A U A HENCE, THE REQUIRED NUMBER OF PROPER SUBSETS OF A U A IS 216 1 ANS (B) 25 IF FOR THREE SETS A, B, C, A ˆ C B ˆ C AND A ˆ CC = B ˆ CC (WHERE CC DENOTES THE COMPLEMENTS OF C), THEN (A) A B (B) B C (C) A C (D) NONE OF THESE SOL A = A ˆ (C ‰ CC) = (A ˆ C) ‰ (A ˆ CC) = (B ˆ C) ‰ (B ˆ CC) = B ˆ (C ‰ CC) = B ANS (A) 26 FOR THREE SETS A, B, C, A B C B C A C C ˆ C ˆ ‰ ˆ ‰ ˆ (A) A (B) B (C) C (D) NONE OF THESE SOL (AC ˆ BC ˆ C) ‰ [(B ‰ C) ˆ C] = [(A ‰ B)C ‰ (A ‰ B)] ˆ C = U ˆ B [U DENOTES A UNIVERSAL SET] = C ANS (C) 27 IF A SET A ADMITS ONLY ONE PROPER SUBSET, THEN N (P(A U A)) IS EQUAL TO (A) SOL ACCORDING TO THE QUESTION A, IS A SINGLETON SET SO, A U A IS ALSO SINGLETON SET HENCE, N(P(A U A)) 21 ANS (B) 28 IF A {1, 2, 3, 4} B {2, 4, 6, 8} C {3, 4, 5, 6} AND U {1, 2, 3, 4, 5, 6, 7, 8, 9}, THEN FIND: (A) AC (B) (A ˆ C)C (C) (AC)C (D) (B – C)C SOL WE HAVE (A) A U A {5,6,7,8,9} C (B) A ˆ C {3, 4} Ÿ (A ˆ C)C = U – (A ˆ C) = {1, 2, 5, 6, 7, 8, 9} (C) (AC)C = U – AC = U – {5, 6, 7, 8, 9} = {1, 2, 3, 4} = A (D) B C {2, 8} Ÿ (B – C)C = U – (B – C) = U – {2, 8} = {1, 3, 4, 5, 6, 7, 9} 29 IF A {1, 3, 5, 7, 9, 11, 13, 15, 17}, B {2, 4, 6, , 18} AND N IS THE UNIVERSAL SET, THEN FIND A A B B C ‰ ^ ‰ ˆ C` SOL WE HAVE (A ‰ B) ˆ BC = A ? AC ‰ {(A ‰ B) ˆ BC} = AC ‰ A = N 30 IF BC AC, SHOW THAT A B SOL WE HAVE BC AC LET X A X A X B [ B A ] Ÿ C Ÿ C C C ' X B Ÿ A B Ÿ 31 CONSIDER THE SET OF ALL DETERMINANTS OF ORDER 3 WITH ENTRIES 0 AND 1 ONLY LET B BE SUBSET OF A CONSISTING OF ALL DETERMINANTS WITH VALUE 1 LET C BE THE SUBSET OF THE SET OF ALL DETERMINANTS WITH VALUE 1 THEN (A) C IS EMPTY (B) B HAS AS MANY ELEMENTS AS C (C) A B ‰ C (D) B HAS TWICE AS MANY ELEMENTS AS C [IIT, 1981] SOL WE KNOW THAT THE INTERCHANGE OF TWO ADJACENT ROWS (OR COLUMNS) CHANGES THE VALUE OF A DETERMINANT ONLY IN SIGN AND NOT IN MAGNITUDE HENCE, CORRESPONDING TO EVERY ELEMENT ' OF B, THERE IS AN ELEMENT 'C IN C OBTAINED BY INTERCHANGING TWO ADJACENT ROWS (OR COLUMNS) IN ' IT FOLLOWS THAT: N(B) D N(C) SETS, RELATIONS AND FUNCTIONS I E , THE NUMBER OF ELEMENTS IN B LESS THAN OR EQUAL TO THE NUMBER OF ELEMENTS IN C SIMILARLY, N(C) D N(B) HENCE, N(B) N(C) I E , B HAS AS MANY ELEMENTS AS C ANS (B) 32 IF A, B AND C ARE NON-EMPTY SUBSETS OF A SET, THEN (A B) ‰ (B A) EQUALS (A) (A ˆ B) ‰ (A ‰ B) (B) (A ‰ B) (A ˆ B) (C) A (A ˆ B) (C) (A ‰ B) B [CEE (DELHI), 1998] SOL (A B) ‰ (B A) (A ‰ B) (A ˆ B) ANS (B) 33 OUT OF 800 BOYS IN A SCHOOL, 224 PLAYED CRICKET, 240 PLAYED HOCKEY AND 336 PLAYED BASKETBALL, OF THE TOTAL, 64 PLAYED BOTH BASKETBALL AND HOCKEY, 80 PLAYED CRICKET AND BASKETBALL AND 40 PLAYED CRICKET AND HOCKEY; 24 PLAYED ALL THE THREE GAMES THE NUMBER OF BOYS WHO DID NOT PLAY ANY GAME IS (A) [CEE (DELHI), 1995] SOL N(C) 224, N(H) 240, N(B) N(H ˆ B) 64, N(B ˆ C) 80, N(H ˆ C) 40, N(C ˆ H ˆ B) N(CC ˆ HC ˆ BC) N[(C ‰ H ‰ B)C] N(U) N(C ‰ H ‰ B) 800 [N(C) + N(H) + N(B) N(H ˆ B) N(B ˆ C) N(H ˆ C) + N(C ˆ H ˆ B)] 800 [224 + 240 + 336 64 80 40 + 24] 800 [824 184] 800 824 + = 984 824 ANS (B) 34 LET A N {AX: X N} AND B N ˆ C N D N, WHERE B, C N ARE RELATIVELY PRIME, THEN (A) C B D (B) B C D (C) D B C (D) NONE OF THESE [CEE (DELHI), 1999] SOL WE HAVE B N THE SET OF +VE INTEGRAL MULTIPLES OF B C N THE SET OF +VE INTEGRAL MULTIPLES OF C ? B N ˆ C N THE SET OF +VE INTEGRAL MULTIPLES OF B C = B C N ? D N B C N ?D B C ANS (C) 35 LET A AND B BE TWO NON-EMPTY SUBSETS OF A SET X SUCH THAT A IS NOT A SUBSET OF B THEN (A) A IS SUBSET OF THE COMPLEMENT OF B (B) B IS A SUBSET OF A (C) A AND B ARE DISJOINT (D) A AND THE COMPLEMENT OF B ARE NON- DISJOINT [UPSEAT, 1995] SOL SINCE A Œ B, X A SUCH THAT X B THEN X BC A B ? ˆ C Z I ANS (D) 36 TWO FINITE SETS HAVE M AND N ELEMENTS, THEN TOTAL NUMBER OF SUB-SETS OF THE FIRST SET IS MORE THAN THE TOTAL NUMBER OF SUBSETS OF THE SECOND THE VALUES OF M AND N ARE (A) 7, (B) 6, (C) 5, (D) 8, [UPSEAT, 1991] SOL SINCE THE TWO FINITE SETS HAVE M AND N ELEMENTS, NUMBER OF SUBSETS OF THESE SETS WILL BE 2M AND 2N RESPECTIVELY ACCORDING TO THE QUESTION, 2M 2N PUTTING M 6, N 3, WE GET 26 23 56 OR 64 8 ANS (B) 37 THE NUMBER OF NON-EMPTY SUBSETS OF THE SET {1, 2, 3, 4} IS (A) [CET (KARNATAKA), 1997] SETS, RELATIONS AND FUNCTIONS SOL NUMBER OF NON-EMPTY SUBSETS OF {1, 2, 3, 4} = 24 1 16 1 ANS (A) 38 IF THE SET A HAS P ELEMENTS AND B HAS Q ELEMENTS, THEN THE NUMBER OF ELEMENTS IN A U B IS (A) P + Q + (D) P + Q [CET (KARNATAKA), 1999] SOL AS A HAS P ELEMENTS AND B HAS Q ELEMENTS, SO A U B HAS PQ ELEMENTS ANS (B) 39 IF TWO FINITE SETS HAVE M AND N ELEMENTS, THEN TOTAL NUMBER OF SUBSETS OF THE FIRST SET IS 56 MORE THAN THE TOTAL NUMBER OF SUBSETS OF THE SECOND THE VALUE OF M AND N ARE (A) 7, [UPSEAT, 1991] SOL SINCE THE TWO FINITE SETS HAVE M AND N ELEMENTS, SO NUMBER OF SUBSETS OF THESE SETS WILL BE 2M AND 2N RESPECTIVELY ACCORDING TO THE QUESTION, 2M 2N PUTTING M 6, N 3, WE GET 26 23 OR 64 8 ANS (B) 40 IF SETS A AND B ARE DEFINED AS A {(X, Y): Y EX, X R} AND B { (X, Y): Y X, X R}, THEN (A) B A (B) A B (C) A ˆ B I (D) A ‰ B A SOL SINCE X X X Y E X , THEREFORE X E X X R ! SO THAT THE TWO CURVES GIVEN BY Y EX AND Y X DO NOT INTERSECT IN ANY POINT HENCE, THERE IS NO COMMON POINT, SO THAT A ˆ B I ANS (C) 41 LET A {1, 2, 3,,10} AND B {1, 2, 3, 4, 5}, THEN THE NUMBER OF SUBSETS C OF A SUCH THAT B ' C {2} IS (A) (C) 24 (D) NONE OF THESE SOL B ' C (B C) ‰ (C B) Ÿ {2} (B C) ‰ (C B) Ÿ C B I AND B C {2} Ÿ C Ž B BUT B {1, 2, 3, 4, 5} ? C CONTAINS THE ELEMENTS 1, 3, 4, Ÿ C {1, 3, 4, 5} HENCE, THE NUMBER OF SUBSETS C OF A IS (24 1) ANS (C) 42 IF A {1, 2, 3, 4}, B {2, 3, 5, 7}, C {3, 6, 8, 9} AND D {2, 4, 8, 10}, THEN (A ' B) ' (C ' D) IS EQUAL TO (A) {1, 5, 6} (B) {4, 8} (C) {2, 3} (D) NONE OF THESE SOL HERE, A ' B {1, 4, 5, 7} AND C ' D {3, 6, 9, 2, 4, 10} ? (A ' B) ' (C ' D) {1, 5, 7, 3, 6, 9, 2, 10} {1, 2, 3, 5, 6, 7, 9, 10} ANS (D) 43 LET A AND B BE TWO SETS SUCH THAT A HAS ELEMENTS AND B HAS 15 ELEMENTS IF A ‰ B HAS 20 ELEMENTS, THEN THE NUMBER OF ELEMENTS OF A ˆ B IS EQUAL TO (A) SOL N(A ‰ B) N(A) + N(B) N(A ˆ B) Ÿ 20 10 + 15 N (A ˆ B) Ÿ N (A ˆ B) 25 20 ANS (B) 44 IF A {X: X IS A PRIME FACTOR OF 12} AND B {1, 2, 3}, THEN (A) A B (B) B Ž A (C) N (A) (D) A ˆ B {1, 3} SOL HERE, A {2, 3} [SINCE 2, ARE THE PRIME FACTORS OF 12] SETS, RELATIONS AND FUNCTIONS B {1, 2, 3} ? N (A) ANS (C) 45 IN A CLASS OF 100 STUDENTS, 55 HAVE PASSED IN MATHS AND 67 HAVE PASSED IN PHYSICS IF EACH STUDENT HAS PASSED IN AT LEAST ONE OF THE TWO SUBJECTS, THEN THE NUMBER OF STUDENTS WHO HAVE PASSED IN MATHS ONLY IS (A) (D) NONE OF THESE SOL HERE, N(M) N(P) NOW, N(M ‰ P) N(M) + N(P) – N(M ˆ P) Ÿ 100 55 + 67 N(M ˆ P) Ÿ N(M ˆ P) 122 100 HENCE, THE NUMBER OF STUDENTS WHO HAVE PASSED ONLY IN MATHEMATICS N(M) N(M ˆ P) 55 22 ANS (B) 46 IF I DENOTES THE NULL SET, THEN (A) I {{I}, {I, {I}}} (B) {I} Ž {I, {I}, {I, {I}}} (C) {I, {I}} Ž {{I}, {I, {I}}} (D) NONE OF THESE SOL {I} CONSISTS OF EXACTLY ONE ELEMENT VIZ , I, AND I IS ALSO SEEN TO BE AN ELEMENT OF THE SET {I, {I}, {I, {I}}} ANS (B) 47 IF A {1, 2, 5}, B {2, 4} C {1, 5, 7}, THEN (A) A \ (B ‰ C) (A \ B) ‰ (A \ C) (B) A ˆ (B ‰ C) A (C) A ‰ (B ˆ C) C (D) NONE OF THESE SOL B ‰ C {1, 2, 4, 5, 7} AND A ˆ (B ‰ C) {1, 2, 5} A ANS (B) MULTIPLE CHOICE QUESTIONS 48 IN A GROUP OF 1000 PEOPLE THERE ARE PEOPLE WHO CAN SPEAK HINDI AND 400 PEOPLE WHO CAN SPEAK ENGLISH THEN NUMBER OF PERSON WHO CAN SPEAK HINDI ONLY IS (A) (D) NONE OF THESE SOL HERE, N(H ‰ E) 1000, N(H) 750, N(E) 400 USING N(H ‰ E) N(H) + N(E) N(H ˆ E) 1000 750 + 400 N(H ˆ E) Ÿ N(H ˆ E) 750 + 400 1150 1000 150 NUMBER OF PEOPLE WHO CAN SPEAK HINDI ONLY N(H ˆ EC) N(H) N(H ˆ E) 750 150 ANS (C) 49 IF X {4N 3N 1: X N} AND Y {9 (N 1): N} PROVE THAT X Y [IIT, 1971] SOL PUTTING N 1, 2, 3, X {0, 9, 54, } Y {0, 9, 18, CLEARLY X Y 50 LET ^ ` A : 2 COS SIN T T T D AND ½ D D ® ¾ ¯ ¿ S S T T FIND A ˆ B [IIT, 1976] SOL ^ ` T T T D ^ ` 2 SIN SIN T T T D ^ ` : SIN 2SIN T T T D ^ ` : SIN 2SIN T T T T 2SIN SIN ½ § T T T T ® ¾ ¨ ¸ © ¹ ¯ ¿ IT IS POSSIBLE IF SIN T D 0 OR SIN T T SIN WHEN = T T S AND IS GREATER WHEN 5 S S D T D S ALSO SIN T D 0 WHERE S D T D 2S SETS, RELATIONS AND FUNCTIONS A : OR S S ½ ? T D T D S D T D S ® ¾ ¯ ¿ B S S ½ T D T D ® ¾ ¯ ¿ A B : OR S S S ½ ? ˆ T D T D S D T D ® ¾ ¯ ¿ 51 IF A N {AX: X N} DESCRIBE THE SET 3N ˆ 7N [IIT, 1970] SOL 3N {3X: X N} = {3, 6, 9, 12, 15, } 7N {7X: X N} = {7, 14, 21, 28, 35, } ? 3N ˆ 7N {21, 42, 63, } {21(1), 21(2), 21(3), } {21X: X N} 21N 52 IF A {1, 2, 3, 4}, DEFINE RELATIONS OF A WHICH ARE (A) R T BUT NOT S (B) S, NEITHER R NOT T (C) R, S AND T [IIT, 1970] SOL FOR R, ADD (1, 1), (2, 2), (3, 3) FOR S, ADD (2, 1), (3, 2) FOR T, ADD (1, 3), (3, 1) THUS THE MINIMUM NUMBER OF ORDERED PAIRS TO BE ADDED IS 7 53 LET A AND B BE TWO DISJOINT SUBSETS OF A UNIVERSAL SET U THEN (A B) B = ‰ ˆ C (A) A (B) B (C) I (D) NONE OF THESE SOL (A ‰ BC) ‰ (B ‰ BC) = A I = A ' A AND B ARE DISJOINT ANS (A) 54 LET A {X: X R, X T 2} AND B {X: X R, X < 4}, THEN A ˆ B (A) {X: X R, 2 < X < 4} (B) {X: X R, 2 D X < 4} (C) B (D) NONE OF THESE SOL TRY YOURSELF ANS (B) 55 A SET CONTAINS N ELEMENTS THE POWER SET CONTAINS (A) N ELEMENTS (B) 2N ELEMENTS (C) N2 ELEMENTS (D) NONE OF THESE [CET, 1992] SOL SINCE A SET CONTAINS N ELEMENTS, THEREFORE THE POWER SET CONTAINS 2N ELEMENTS ANS (B) 56 THE NUMBER OF BIJECTIVE FUNCTION FROM SET A TO ITSELF WHEN A CONTAINS 106 ELEMENTS IS (A) (C) (106)! (D) (106)! ANS (C) 57 TWO FINITE SETS HAVE M AND N ELEMENTS THE TOTAL NUMBER OF SUBSETS OF THE FIRST SET IS MORE THAN THE TOTAL NUMBER OF SUBSETS OF THE SECOND SET THE VALUES OF M AND N ARE (A) 7, [MNR, 1991] SOL GIVEN THAT 2M 2N BY TRIAL, M 6 AND N WE GET 26 23 64 8 56 ANS (B) 58 LET S {1, 2, 3, 4} THE TOTAL NUMBER OF UNORDERED PAIRS OF DISJOINT SUBSETS OF S IS EQUAL TO (A) [IIT, 2010] SOL S {1, 2, 3, 4} EACH ELEMENT CAN BE PUT IN 3 WAYS EITHER IN SUBSETS OR WE DO NOT PUT IN ANY SUBSET ? TOTAL NUMBER OF ORDERED PAIRS U U U 81 = 40 + 1 = ANS (C) SETS, RELATIONS AND FUNCTIONS 59 LET X {1, 2, 3, 4, 5} THE NUMBER OF DIFFERENT ORDERED PAIR (Y, Z) THAT CAN BE FORMED SUCH Y Ž X, Z Ž X AND Y ˆ Z IS EMPTY (A) [AIEEE, 2012] SOL EVERY ELEMENT HAS 3 OPTIONS EITHER SET Y OR SET Z OR NONE THEREFORE, NUMBER OF ORDERED PAIR 35 ANS (D) 60 25 PEOPLE FOR PROGRAMME A, 50 PEOPLE FOR PROGRAMME B, 10 PEOPLE FOR BOTH, SO NUMBER OF EMPLOYEE EMPLOYED ONLY A IS (A) [ORISSA-JEE, 2010] SOL N (A B) N (A) N (A ˆ B) 25 ANS (A) 61 THE SET CONTAINS 5 ELEMENTS, THEN THE NUMBER OF ELEMENTS IN THE POWER SET P(A) IS EQUAL TO (A) [KERALA-ENG, 2011] SOL THE SET CONTAINS 5 ELEMENTS ? THE NUMBER OF ELEMENTS IN THE POWER SET P(A) 25 ANS (B) 62 GIVEN A {X: X IS A ROOT OF X2 1 0} AND