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De Morgan's Law

A subset S of...

Question

A subset S of the set of numbers {2, 3, 4, 5, 6, 7, 8, 9, 10} is said to be good if it has exactly 4 elements and their greatest common divisor is 1. Let ‘n’ be the number of good subsets. Then the units digit of ‘n’ is___

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S = {1, 2, 3, \dots, n}

A = {(a, b) | 1 \leq a, b \leq n} = S \times S

B

A

is said to be a good subset if

(x, x) \in B

x \in S

. Then the number of good subsets of

A

S = {1, 2, 3, . . . . . . . n}

A = {(a, b) | 1 \geq a, b \geq n} = S \times S

B

A

is said to be a good subset if

(x, x) \in B

x \in S .

Then the number of good subsets of

A

n

be a positive integer. Call a non-empty subset

S

{1, 2, . . ., n}

good, if the arithmetic mean of the elements of

S

, is also an integer. Further let

t_{n}

denote the number of good subsets of

{1, 2, . . ., n}

t_{n}

n

are both odd or both even.

5

n

be the number of ways of partitioning the set into

2

or more non-empty subsets. Then

n

n

be a natural number and

X = {1, 2, . . . . . ., n}

A

B

X,

A Δ B

to be the set of all those elements of

X

which belong to exactly one of

A

B .

F

be a collection of subsets of

X

such that for any two distinct elements

A

B

F

A Δ B

has at least two elements. Show that

F

2^{n - 1}

elements. Find all such collections

F

2^{n - 1}

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