SET THEORY
Introduction
Set: A well defined collection of objects is called a set.
Finite set: a set of finite elements.
Null set: set with no elements.
Singleton set: set with one element only.
Equal set: each element of one set is in the other and vice versa.
Subset: Every element of A is element of B. Denote it as A⊆B A={1,2} and B={1,2,3} or B={1,2}. Proper subset is if there is at least one element in B but not in A but all elements in A are in B. A⊂B. E.g: A = {1,2,3} and B={1,2,3,4}.
Union of sets: A∪B = {all elements of A and B but no duplicates};
Intersection of sets: A∩B = {common elements of A and B but no duplicates}
Difference of sets: A-B = {set of all elements of A not in B}
Complement of set: If A is the subset of X then the a set of all elements in X but not in A is Ac
Note: A set of ‘n’ elements can have total 2n
subsets.
Set Theory
- Set Theory is an important concept of mathematics which is often asked in aptitude exams. There are two types of questions in this chapter:
1.Numerical questions on set theory based on venn diagrams
2.Logical questions based on set theory
- Let us first take a look at some standard theoretical inputs related to set theory
- Figure 1: Refers to the situation where there are two attributes A and B. (Let’s say A refers to people who passed in Physics and B refers to people who passed in Chemistry.) Then the shaded area shows the people who passed both in Physics and Chemistry.
- In mathematical terms, the situation is represented as: Total number of people who passed at least 1 subject = A + B – (A ∩ B)
- Figure 2: Refers to the situation where there are three attributes being measured. In the figure below, we are talking about people who passed Physics, Chemistry and/or Mathematics.
- In the above figure, the following explain the respective areas:
- Area 1: People who passed in Physics only
- Area 2: People who passed in Physics and Chemistry only (in other words-people who passed Physics and Chemistry but not Mathematics)
- Area 3: People who passed Chemistry only
- Area 4: People who passed Chemistry and Mathematics only (also, can be described as people who passed Chemistry and Mathematics but not Physics)
- Area 5: People who passed Physics and Mathematics only (also, can be described as people who passed Physics and Mathematics but not Chemistry)
- Area 6: People who passed Physics, Chemistry and Mathematics
- Area 7: People who passed Mathematics only
- Area 8: People who passed in no subjects
- Also take note of the following language which there is normally confusion about:
- People passing Physics and Chemistry-Represented by the sum of areas 2 and 6
- People passing Physics and Maths-Represented by the sum of areas 5 and 6
- People passing Chemistry and Maths-Represented by the sum of areas 4 and 6
- People passing Physics-Represented by the sum of the areas 1, 2, 5 and 6
- In mathematical terms, this means: Total number of people who passed at least 1 subject = P + C + M – (P ∩ C) – (P ∩ M) – (C ∩ M) + (P ∩ C ∩ M)
Solved examples – Set theory
Q.
At the birthday party of Sherry, a baby boy, 40 persons chose to kiss him and 25 chose to shake hands with him. 10 persons chose to both kiss him and shake hands with him. How many persons turned out at the party?
35
45
55
65
Q.
In an examination 43% passed in Math, 52% passed in Physics and 52% passed in Chemistry. Only 8% students passed in all the three. 14% passed in Math and Physics and 21% passed in Math and Chemistry and 20% passed in Physics and Chemistry. Number of students who took the exam is 200. Let Set P, Set C and Set M denotes the students who passed in Physics, Chemistry and Math respectively. Then How many students passed in Math only?
1.32
2.42
3.45
4.52
Q.
In an examination 43% passed in Math, 52% passed in Physics and 52% passed in Chemistry. Only 8% students passed in all the three. 14% passed in Math and Physics and 21% passed in Math and Chemistry and 20% passed in Physics and Chemistry. Number of students who took the exam is 200. Let Set P, Set C and Set M denotes the students who passed in Physics, Chemistry and Math respectively. Find the ratio of students passing in Math only to the students passing in Chemistry only?
1.16:19
2.6:1
3.16:9
4.1:19
Q.
In an examination 43% passed in Math, 52% passed in Physics and 52% passed in Chemistry. Only 8% students passed in all the three. 14% passed in Math and Physics and 21% passed in Math and Chemistry and 20% passed in Physics and Chemistry. Number of students who took the exam is 200. Let Set P, Set C and Set M denotes the students who passed in Physics, Chemistry and Math respectively. What is the ratio of the number of students passing in Physics only to the students passing in either Physics or Chemistry or both?
1.26:84
2.26:8
3.2:84
4.52:84
Q.
In an examination 43% passed in Math, 52% passed in Physics and 52% passed in Chemistry. Only 8% students passed in all the three. 14% passed in Math and Physics and 21% passed in Math and Chemistry and 20% passed in Physics and Chemistry. Number of students who took the exam is 200. Let Set P, Set C and Set M denotes the students who passed in Physics, Chemistry and Math respectively. A student is declared pass in the exam only if he/she clears at least two subjects. The number of students who were declared passed in this exam is?
1.32
2.42
3.45
4.78
Q.
In the Mindworkzz club all the members participate either in the Tambola or the Fete. 320 participate in the Fete, 350 participate in the Tambola and 220 participate in both. How many members does the club have?
1.410
2.450
3.440
4.380
Q.
There are 20000 people living in Defence Colony, Gurgaon. Out of them 9000 subscribe to Star TV Network and 12000 to Zee TV Network. If 4000 subscribe to both, how many do not subscribe to any of the two?
1.5000
2.2000
3.3000
4.1000
Q.
Last year, there were 3 sections in the Catalyst, a mock CAT paper. Out of them 33 students cleared the cut-off in Section 1, 34 students cleared the cut-off in Section 2 and 32 cleared the cut-off in Section 3. 10 students cleared the cut-off in Section 1 and Section 2, 9 cleared the cut-off in Section 2 and Section 3, 8 cleared the cut-off in Section 1 and Section 3. The number of people who cleared each section alone was equal and was 21 for each section. How many cleared all the three sections?
1.3
2.5
3.6
4.7
Q.
Last year, there were 3 sections in the Catalyst, a mock CAT paper. Out of them 33 students cleared the cut-off in Section 1, 34 students cleared the cut-off in Section 2 and 32 cleared the cut-off in Section 3. 10 students cleared the cut-off in Section 1 and Section 2, 9 cleared the cut-off in Section 2 and Section 3, 8 cleared the cut-off in Section 1 and Section 3. The number of people who cleared each section alone was equal and was 21 for each section. How many cleared only one of the three sections?
1.33
2.35
3.63
4.74