Question

Find all the partitions of the set A={ a , b , c , d }.

Answer 1

Answer:

9353982311

fashiondeal,,''customer,,''sarvice number,,9353982311+*7318844196

Answer 2

Given data:

A set A = {a, b, c, d}

To find:

All the partitions of the set A

Concept to be used:

Partition of a set. Let $S$ be a non-empty set. A family of non-empty subsets $\{S_{\alpha}:\alpha\in I\}$, $I$ being the index set, is said to form a partition of $S$ if

(i) ${\cup}_{\alpha\in I}=S$

and (ii) $S_{\alpha}\cap S_{\beta}=\phi$ for $\alpha,\beta\in I$ and $\alpha\neq\beta$

Step-by-step explanation:

All the partitions of the set A are

{{a}, {b}, {c}, {d}}

{{a, b}, {c}, {d}}

{{a, c}, {b}, {d}}

{{a, d}, {b}, {c}}

{{a}, {b, c}, {d}}

{{a}, {b, d}, {c}}

{{a}, {b}, {c, d}}

{{a, b, c}, {d}}

{{a, b, d}, {c}}

{{a, c, d}, {b}}

{{a}, {b, c, d}}

{{a, b}, {c, d}}

{{a, c}, {b, d}}

{{a, d}, {b, c}}

{{a, b, c, d}}

It is clear all the above partitions satisfy the condition for partitions of a set.

NOTE:

There are 15 partitions for a set containing 4 elements. This can be deduced from the concept of Bell Number $B_{n}$; where $B_{4}=15$.

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