Define ceiling and floor function. Why do we need Inclusion – Exclusion principle? Make it clear with suitable example.

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Let x be a real numbers. The floor function of x, denoted by ⌊x⌋ or floor(x), is defined to be the gratest integer that is less than or equal to x.

For example: ⌊1.4⌋ = 1 and ⌊5⌋ = 5

The ceiling function of x, denoted by ⌈x⌉ or ceil(x), is defined to be the greatest integer that is greater than or equal to x

For example: ⌈5⌉ = 5 and ⌈1.5⌉ = 2

Inclusive-Exclusive Principle:

It is a counting technique which generalizes the familiar method of obtaining the number of elements in the union of two or more finite steps.

Symbolically represented as

|A∪B| = |A| + |B| – |A∩B|

|A ∪ B ∪ C| = |A| + |B| + |C| – |A∩B| – |B∩C| – |A∩C| + |A∩B∩C|

We need inclusion-exclusion principle to distinguish the number of distinct elements which exist in a set.

An example to distinguish the number of integers in {1, 2, 3, 4, ……………….., 100} that are divisible by 2, 3, or 5 is

Let A = {an integer that is divisible by 2}

∴ |A| = 50

B = {an integer that is divisible by 3}

∴ |B| = 33

C= {an integer that is divisible by 5}

∴ |C| = 20

Now, A ∩ B = { divisible by 2 and 3 } = 16

Similarly,

|B ∩ C|  = 6

|A ∩ C| = 10

|A ∩ B ∩ C| = 3

By inclusion exclusion principle,

|A ∪ B ∪ C| = |A| + |B| + |C| – |A∩B| – |B∩C| – |A∩C| + |A∩B∩C|

=50 + 33 + 20 – 16 – 10 – 6  + 3

= 74

Here, we distinguish the no of integers that are divisible by 2,3 or 5 using inclusion-exclusion principle.

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