Mathematics Village

Day of the Week:

January has 31 days. It means that every date in February will be 3 days later than the same date in January(28 is 4 weeks exactly). The below table is calculated in such a way. Remember this table which will help you to calculate.

January	0
February	3
March	3
April	6
May	1
June	4
July	6
August	2
September	5
October	0
November	3
December	5

Step1: Ask for the Date. Ex: 23^rd June 1986
Step2: Number of the month on the list, June is 4.
Step3: Take the date of the month, that is 23
Step4: Take the last 2 digits of the year, that is 86.
Step5: Find out the number of leap years. Divide the last 2 digits of the year by 4, 86 divide by 4 is 21.
Step6: Now add all the 4 numbers: 4 + 23 + 86 + 21 = 134.
Step7: Divide 134 by 7 = 19 remainder 1.
The reminder tells you the day.

Sunday	0
Monday	1
Tuesday	2
Wednesday	3
Thursday	4
Friday	5
Saturday	6

Answer: Monday

Set Theory formulas

Set Theory is a branch of mathematics which deals with the study of sets or the collection of similar objects.
Set theory is one of the most fundamental branch of mathematics.

Set Theory Formulas:

Notations used in set theory formulas:

$n(A)$ – Cardinal number of set A.

$n_o(A)$ – cardinality of set A.

$\bar{A} = A^c$ – Complement of set A.

$U$ – universal set

$A \subset B$ – Set A is proper subset of subset of set B.

$A \subseteq B$ – Set A is subset of set B.

$\phi$ – Null set.

$a \in A$ – element “a” belongs to set A.

$A \cup B$ – Union of set A and set B.

$A \cap B$ – Intersection of set A and set B.

Formulas:

For a group of two sets:

1. If A and B are overlapping set, $n(A \cup B) = n(A) + n(B) - n(A \cap B)$

2. If A and B are disjoint set, $n(A \cup B) = n(A) + n(B)$

3. $n(A) = n(A \cup B) +n(A \cap B) - n(B)$

4. $n(A \cap B) = n(A) +n(B) - n(A \cup B)$

5. $n(B) = n(A \cup B) +n(A \cap B) - n(A)$

6. $n(U) = n(A) + n(B) - n(A \cap B) + n((A \cup B)^c)$

7. $n((A \cup B)^c) = n(U) + n(A \cap B) - n(A) - n(B)$

8. $n(A \cup B) = n(A - B) + n(B - A) + n(A \cap B)$

9. $n(A - B) = n(A \cup B) - n(B)$

10. $n(A - B) = n(A) - n(A \cap B)$

11. $n(A^c) = n(U) - n(A)$

For a group of three sets:
1. $n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(A \cap C) - n(B \cap C) + n(A \cap B \cap C$

2. $n(A \cup B \cup C) = n(U) - n((A \cup B \cup C)^c)$

3. $n(A \cap B \cap C) = n(A \cup B \cup C)+n(A \cap B)+n(A \cap C)+n(B \cap C) - n(A) - n(B) - n(C)$

4. $n_0(A) = n(A)-n(A \cap B)-n(A \cap C)+n(A \cap B \cap C)$

5. $n_0(B) = n(B)-n(A \cap B)-n(B \cap C)+n(A \cap B \cap C)$

6. $n_0(C) = n(C)-n(A \cap C)-n(B \cap C)+n(A \cap B \cap C)$

7. $n(A \cap B only) = n(A \cap B)-n(A \cap B \cap C)$

8. $n(B \cap C only) = n(B \cap C)-n(A \cap B \cap C)$

9. $n(A \cap C only) = n(A \cap C)-n(A \cap B \cap C)$

Centroid of a Triangle

In mathematics and physics, the centroid or geometric center of a two-dimensional region is, informally, the point at which a cardboard cut-out of the region could be perfectly balanced on the tip of a pencil (assuming uniform density and a uniform gravitational field). Formally, the centroid of a plane figure or two-dimensional shape is the arithmetic mean ("average") position of all the points in the shape. The definition extends to any object in n-dimensional space: its centroid is the mean position of all the points in all of the coordinate directions.While in geometry the term barycenter is a synonym for "centroid", in physics "barycenter" may also mean the physical center of mass or the center of gravity, depending on the context. The center of mass (and center of gravity in a uniform gravitational field) is the arithmetic mean of all points weighted by the local density or specific weight. If a physical object has uniform density, then its center of mass is the same as the centroid of its shape. In geography, the centroid of a radial projection of a region of the Earth's surface to sea level is known as the region's geographical center. The point where the three medians of the triangle intersect. The 'center of gravity' of the triangle One of a triangle's points of concurrency.Refer to the figure . Imagine you have a triangular metal plate, and try and balance it on a point - say a pencil tip. Once you have found the point at which it will balance, that is the centroid.The centroid of a triangle is the point through which all the mass of a triangular plate seems to act. Also known as its 'center of gravity' , 'center of mass' , or barycenter.A fascinating fact is that the centroid is the point where the triangle's medians intersect. See medians of a triangle for more information. In the diagram above, the medians of the triangle are shown as dotted blue lines.

Centroid facts
•The centroid is always inside the triangle
•Each median divides the triangle into two smaller triangles of equal area.
•The centroid is exactly two-thirds the way along each median.Put another way, the centroid divides each median into two segments whose lengths are in the ratio 2:1, with the longest one nearest the vertex. These lengths are shown on the one of the medians in the figure at the top of the page so you can verify this property for yourself.