## The need for a Universal Set: Basic Set Theory

For any two sets A and B, we define $A \setminus B=\{ x: x \in A \hspace{0.1in} \textup{and} \hspace{0.1in} x \notin B \}$.

$A\setminus B$ is the set of all those elements of A which are not in B. If E stands for the set of those persons who can speak English and H for the set of persons who can speak Hindi, then $E\setminus H$ would stand for the set of all those persons who can speak English but cannot speak Hindi. Similarly, $H\setminus E$ would stand for the set of all  those persons who can speak Hindi but not English. The sets $A\setminus B$ and

$B\setminus A$ are called the set of differences. In this context, there is an interesting concept called the symmetric difference of two sets. For sets A and B, the set $(A\setminus B) \bigcup (B\setminus A)= A \triangle B$.

For the sets E and H described above, $E\triangle H$ stands for the set of all those persons who can either speak English or speak Hindi but not both. This is the case of exclusive or in digital logic or mathematical logic.

Exercise:

In an examination of three subjects physics, chemistry and mathematics, let P stand for the set of those candidates who passed in physics, C for the set of those candidates who passed in chemistry and M for the set of those who passed in mathematics.

Express the following sets by using the symbols of union, intersection and difference.

(a) The set of all those candidates who  passed in mathematics only.

(b) The set of all those who passed in mathematics and physics but not in chemistry.

(c) The set of those candidates who failed in one subject only.

(d) The set of those candidates who passed in either physics or chemistry but not in mathematics (we or not exclusive or here)

(e) The set of those candidates who passed in all the subjects.

(f) The set of all those candidates who passed at least in two subjects.

In this exercise, if we want to represent the set of those candidates who failed at least in one subject, what do we do? An obvious problem in expressing this as a set if that we have not specified the set of all the candidates who  took the examination. Similarly, if we are talking of the set of non-Bengalis we must be clear in which context we are talking? Are we talking of all the people of the world or only about Indian citizens? But when we say ‘not all birds can fly” we are talking of birds only. We are not talking of the other animals who cannot fly nor are we talking of insects which we can fly. Similarly, suppose somebody said “none other than Gandhi could do this feat” we are obviously talking in terms of all human beings. We can find many such examples where it is important to mention the context in which we are talking. In other words, we specify a set in terms of subsets of a set about which we are talking. This specified set is sometimes called the “the universe of discourse” or the universal set, U. Though we are using the article “the” we are talking of “the universal set” in a particular context.

If is our universal set and $A \subset U$, then the set   $U \setminus A=\{ x: x \in U \hspace{0.1in} \textup{and} \hspace{0.1in} x \notin A \}$ is called the complement of A. It is denoted by $A^{'}$. (Strictly speaking, we should call

$U \setminus A$ as the complement of A with respect to U). In case, U is our universe of discourse, we can legitimately claim $U^{'} = \phi$.

More later,

Nalin Pithwa

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