In a group of nine people, one person knows exactly two others, two people know exactly 5 others, and another 2 people know exactly 6 others. show that there are three people who all know each other

1, 2 and 5 all know each other.

Step-by-step explanation:

I will denote the people by numbers: 1 2 3 4 5 6 7 8 9

The following information is given in the question.

1 knows two others

2 knows five others

3 knows five others

4 knows six others

5 knows six others

Let's assume the worst case: They all know different persons, so it is difficult to find three people who all know each other.

1 knows 2 and 5

2 knows 1 3 4 5 6

3 knows 9 8 7 6 2

By now, 2, 5 and 6 are known to two people, and others are known to just one person.

4 knows 9 8 7 6 5 3

5 knows 1 2 4 7 8 9

As a result 1, 2 and 5 all know each other.

1 knows 2 and 5

2 knows 1 and 5

5 knows 1 and 2

The reason for this is that there are three groups:

1, (2,3) and (4,5)

As much as we try to separate the people that they know there are nine different persons and at least three of them will be spread to these groups in common.

the system of inequity is

step-by-step explanation:

from the given graph it is noticed that we have two related lines. one is solid and another is dashed.

if a line passing through two points, then the equation of line is

the solid line passing through the points (0,4) and (4,0).

the related equation of solid line is .

similarly, find the related equation of dashed line. the dashed liune passing through the points (0,0) and (3,1).

the related equation of dashed line is

from the given graph it is noticed that the point (0,1) lies on the shaded region, therefore the point (0,1) must be satisfied by the related equations.

check the related equations by the point (0,1).

this statement is true if the sign of inequality is < instead of equality. since it is an solid line therefore the sign of inequality will be . the required inequality is

similarly, check the second related equation.

it is a dashed line, so this statement is true if the sign of inequality is > . the required inequality is