# How to find the distinct equivalence classes for the set of all bit strings of length 5

Let B denote the set of all bit strings of length 5, $b_1,b_2,b_3,b_4,b_5$. Define a relation R on B by: two bit strings are related by R if and only if they both have bits $b_1$ the same and both have the bits $b_5$ the same.

(a) List all the elements of the equivalence class [10010].

(b) How many distinct equivalence classes are there? List them.

So we never went over bit strings in class and I'm trying to apply the same concepts as a similar question that had integers and ordered pairs.

For part (a), the elements of the equivalence class [10010] is just 0 and 1?

For part (b), the distinct equivalence classes are all of the variations of the 5 bit strings where bits $b_1$ and $b_5$ are the same? for example, [10010] and [11110]? How would I go about determining the exact number of distinct equivalent classes?

Any help is appreciated,

• Well, can you give a single string equivalent to $10010$ other than itself? – lulu Sep 7 '17 at 23:11
• I don't understand your answer "$0$ and $1$"...neither of those are strings of length $5$. – lulu Sep 7 '17 at 23:12
• Every string with $b_1 = 1$ and $b_5 = 0$ is in the equivalence class of 10010 – Namaste Sep 7 '17 at 23:13
• I guess I'm confused by what the question means by the word element. An element of the equivalence class. – Jamie Sep 7 '17 at 23:13
• Ah, thanks. so every string with [1 _ _ _ 0] and filling in the blanks with variations of 0 or 1 is an element of [10010]. – Jamie Sep 7 '17 at 23:18