Logical Reasoning

- 1. Introduction
- 2. Basics of Set Theory
- 3. Venn Diagrams
- 4. Operation of Sets
- 5. Problem Solving Using Venn Diagrams
- 6. Examples

Venn diagrams are an efficient way of representing and analyzing sets and performing set operations. As such, the usage of Venn diagrams is just the elaboration of a solving technique. Problems that are solved using Venn diagrams are essentially problems based on sets and set operations. Thus, before we move on to understanding Venn diagrams, we first need to understand the concept of a set.

This section defines the terms set, universal set and subset – three terms that are essential in understanding set theory.

A well-defined collection of objects is called a set. Each member of a set is called an element. All elements of a set follow a certain rule and share a common property amongst them.

For example, the teams of the countries who qualified for the Quarterfinals of the 2006 Football World Cup can constitute a set. There will be 8 members in this set and the rule that is common to them is that all of them are teams that have reached the quarterfinals of the 2006 Football World Cup.

To understand how sets are denoted, take another example. Consider the set of vowels in the English alphabet. Let this set be represented by the letter *V*. This set contains 5 elements – *a*, *e*, *i*, *o*, *u*. This can also be written as:

$V = \{a, e, i, o, u\}$

The following two conditions should be taken care of while writing a set –

A set must be denoted by a capital letter.

The elements of the set must be denoted in small letters.

Following are certain examples of sets –

$A = \{1, 2, 3, 4, 5\}$

$B = \{a, b, c, d, e\}$

In the above examples, *A* and *B* are sets while 1, 2, 3, 4, 5 and *a*, *b*, *c*, *d*, *e* are the elements belonging to these sets respectively.

The symbol $\epsilon$ is used to denote that an element belongs* *to a set.

Certain sets are used very frequently, like the set of all natural numbers. Following are certain standard letters used for particular sets –

*N* = Set of all natural numbers = {1, 2, 3, 4 …}

*I* = Set of all integers = {…, -3, -2, -1, 0, 1, 2, 3 …}

*Q* = Set of all rational numbers

*R* = Set of all real numbers

*P* = Set of all prime numbers

*C* = Set of all complex numbers

A set that contains all the elements and sets in a given scenario is called a Universal Set (*U*).

In our previous example of Football teams, the universal set can be considered as the set of all international teams that play Football.

For example,

*U* = Set of all numbers

The universal set for the set of all numbers is *U* = {$-\infty ... \infty$}. It includes all natural numbers, integers, rational numbers, irrational numbers, etc.

Set *B* is said to be a *subset* of another set *A* when all elements of set *B* are also elements of set *A*. Set *A* is said to be a **superset** of set *B*.

Using the same example, if *A* is the set of all teams that took part in the 2006 Football World Cup and *B* is the set of teams that reached the Quarterfinals, then *B* is said to be a subset of *A*. This simply means that all the teams present in *B* must definitely be present in *A*.

Let us consider another example:

*A* = {1, 2, 3, 4, 5, 6} and *B* = {1, 2, 3}

We can see that every element of set *B* is an element of set *A*.

Hence, we can say that *B* is a subset of *A*. We can also say that *B* is contained in* A*.

*A* is said to be the superset of *B*.

In any given context or scenario all sets are subsets of the Universal set.

REMEMBER:

- A Null Set (a set that has no elements) is a subset of every set.
- Every set is a subset of itself.

Now that we know what sets are, we can look at Venn Diagrams as an alternate way of depicting sets. Venn Diagrams consist of closed shapes, generally circles, which represent sets. The capital letter outside the circle denotes the name of the set while the letters inside the circle denote the elements of the set.

The various operations of sets are represented by partial or complete overlap of these closed figures. Regions of overlap represent elements that are shared by sets.

In practice, sets are generally represented by circles. The universal set is represented by a rectangle that encloses all other sets. Venn Diagrams are generally not drawn to scale.

The above figure is a representation of a Venn diagram. Here each of the circles *A*, *B* and *C* represents a set of elements.

Set *A* has the elements *a*, *d*, *e* and *g.*

*S*et *B* has the elements *b*, *d*, *g* and *f.*

*S*et *C* has the elements *e*, *g*, *f* and *c*.

Both *A* and *B* have the elements *d* and *g*.

Both* B* and *C* have the elements *g* and *f*.

Both *C* and *A* have the elements *e* and *g*.

*A*, *B* and *C* all have the element *g*.

Let us now look at few basic set operations and ways of representing them using Venn diagrams. For understanding these operations, we will use a common example and perform operations on it.

Consider a class of students that form the universal set. Set *A* is the set of all students who were present in the English lecture, while Set *B* is the set of all the students who were present in the History lecture. It is obvious that there were students who were present in both lectures as well as those who were not present in either of the two lectures.

Complement of a set *A* in the given context is the set having all elements that belong to the Universal set but not to *A*.

In our example, the complement of set *A* will be all the students who were absent in the English lecture.

Suppose,

*U* = {*a*, *b*, *c*, *d*, *e*, *f*, *g*, *h*} and *A* = {*a*, *b*, *c*, *d*, *e*},

Then *A*’, or complement of the set *A* = {*f*, *g*, *h*}

The union of two sets *A* and *B* is defined as the set having all the elements which belong to either *A* or *B* or both *A* and *B*.

In our example, the union of sets *A* and *B* will contain all the students who were present in at least one of the two lectures. Only students who did not attend a single lecture will not be considered in the union.

The intersection of sets *A* and *B* is defined as the set having all elements which belong to both *A* and *B*.

In our example, the intersection of *A* and *B* will contain all the students who sat for both, English as well as History lectures.

The difference of two sets *A* and *B*, *A* - *B*, is defined as the set having all the elements of A that do not belong to *B*. Please note that *A* - *B* is not always equal to *B* - *A*.

**Example 1: **In our example, *A - B* will be all the students who have attended only the English lecture and not the history lecture while *B - A* will be all the students who have attended just the History lecture and not the English lecture.

**Example 2: **Let us consider another example. In a room, there are 5 people *a*, *b*, *c*, *d*, *e*. Out of them, *a*, *b* and *c* are men while *d* and *e* are women. Also, *a* and *e* study science while *b*, *c* and *d* study commerce.

$\therefore$ The set of males is:

*M* = {*a*, *b*, *c*}

The set of females is:

*F* = {*d*, *e*}

The set of science students is:

*S* = {*a*, *e*}

The set of commerce students is:

*C* = {*b*, *c*, *d*}

If we wish to find out all female students who have taken science, we need to find out what is common in set *F* and set *S*. This is called an intersection of set *F* and set *S* and is denoted by *F* ∩ *S*.

Here, *F* ∩ *S* = {*e*}

Thus, an intersection of two sets is formed by the elements which are common to both the sets.

Similarly, if we consider sets *M* and *F*, there is no common element between them.

Hence, *M* ∩ *F* = *$\phi$*

Such sets which have no elements in common are called **disjoint sets. **

Now, let us find out those females who have not taken science. Here, we have to check the set *F* and remove all elements of set *S* present in this set. This is called the difference between two sets.

*S* - *F* = {*a*}

Thus, difference of set *A* and set *B* is defined as the set of all elements present in *A* but not in *B*.

*A* - *B* = {*x* | *x* ( *A* and *x* ∉ *B*}

Now, suppose you want to represent a set containing ”either males or commerce students or both”. This would mean taking all the elements from set *M* and set *C* together into one set. This is called the union of set *M* and set *C* and is denoted by *M* $\cup$ *C*.

Thus, *M* $\cup$ *C* = {*a*, *b*, *c*, *d*}

Though *b* and *c* exist in both sets, they are written only once while writing the union. This is because no element is ever written twice while writing a set.

Let us take an example to understand Venn diagrams.

Let *U* be the universal set containing all the natural numbers between 0 and 11.

Hence, *U* = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Let *P* be the set containing all the prime numbers between 0 and 11.

Thus, *P* = {2, 3, 5, 7}

Let *E* be the set containing all the even numbers between 0 and 11.

Hence, *E* = {2, 4, 6, 8, 10}

Hence, *P* $\cup$ *E* = {2, 3, 4, 5, 6, 7, 8, 10}

Also, *P* ∩ *E* = {2}

This can be represented using a Venn diagram in the following manner-

In the above diagram, the universal set is represented by the rectangle while all other sets are represented by circles. The common portion between the two sets *P* and *E* is the intersection of these two sets.

The universal set contains two numbers, 1 and 9, which do not belong to the set of prime or even numbers. Hence, they are written outside both circles but within the universal set.

*n*(*P*) = 4, n(*E*) = 5 and *n*(*P* $\cup$ *E*) = 8

Thus if we observe, *n*(*P* $\cup$ *E*) ≠ *n*(*P*) + *n*(*E*)

This is because the element 2 is included in both *P* and *E* but while taking the union of the two sets, we count it only once.

Therefore, if we subtract *n*(*P* ∩ *E*) from *n*(*P*) + *n*(*E*), then we will get *n*(*P* $\cup$ *E*).

*n*(*P* $\cup$ *E*) = *n*(*P*) + *n*(*E*) - *n*(*P* ∩ *E*)

i.e. *n*(*P* $\cup$ *E*) = 4 + 5 – 1

= 8

To generalize this, for any two sets *A* and *B*,

*n*(*A* $\cup$ *B*) = *n*(*A*) + *n*(*B*) - *n*(*A* ∩ *B*)

Similarly, now let us consider a universal set as follows:

*U* = {1, 2, 3, 4, 5, 6, 7, 8, 9, 15}

Let *A* be the set of all even numbers.

Hence, *A* = {2, 4, 6, 8}

Let *B* be the set of all square numbers.

Hence, *B* = {1, 4, 9}

Let *C* be the set of all composite numbers.

Hence, *C* = {4, 6, 8, 9, 15}

This can be represented using a Venn diagram as follows:

For a three level Venn diagram, the formula is

*n* (*A* $\cup$ *B* $\cup$* C*) = *n* (*A*) + *n* (*B*) + *n* (*C*) - *n* (*A* ∩ *B*) – *n* (*A* ∩ *C*) - *n* (*B* ∩ *C*) + *n* (*A* ∩ *B* ∩ *C*)

So in this case,

*n*(*A* $\cup$ *B* $\cup$ *C*) = 4 + 3 + 5 - 1 - 3 - 2 + 1 = 7

Draw Venn diagrams for:

**(1)** A - *B* – *C*

**(2)** (*A* ∩ *B*) $\cup$ *C*’

**(3)** (*A* $\cup$ *B*) ∩ *C*’

**(1)** *A* - *B* – *C*

**(2)** (*A* ∩ *B*) $\cup$* C*’

**(3)** (*A* $\cup$ *B*) ∩ *C*’

Express the following as Venn diagrams:

**(1)** Every bull is an animal

**(2)** No animal is a bull

Let *A* = Set of Animals, *B* = Set of bulls

**(1)** Every bull is an animal.

**(2)** No animal is a bull

Here, *A* and *B* are disjoint sets.