Wednesday, August 21, 2002

Experience Mathematics #10 -- Sets

A fundamental object in mathematics is a set. You can think of a set as a collection of objects. We are familiar with the set $N$ of natural numbers. The empty set is a set with no elements.

We say that a set $B$ is a subset of a set $A$ if each element of $B$ is also in $A$. Two sets are equal if they are subsets of each other. The empty set is a subset of every set. For example the set of even numbers $\left\{ 2, 4, 6, \dots\right\}$ is a subset of $N$.

Q1. How many subsets does the empty set have? (Hint: There is atleast one subset.)

The members of the set are called its elements.

Let $I(n)$ denote the set containing the first $n$ natural numbers. The set $I(n)$ has n elements.

Q2. List all the subsets of the set $I(1)$. How many subsets does $I(1)$ have?

Q3. List all the subsets of the set $I(2)$. How many subsets does $I(2)$ have?

Q4. List all the subsets of the set $I(3)$. How many subsets does $I(3)$ have?

Suppose you delete $3$ from each subset of $I(3)$. What do you get? Use this idea to do Q5.

Q5. List all the subsets of the set $I(4)$. How many subsets does $I(4)$ have?

Q6. How many subsets does $I(n)$ have? Guess the answer from your experiments above.

If you cannot find a pattern, you must have made a mistake in listing the sets earlier.