Study Definition of Subset

Online gives the definition of subset as the elements of subset are contained by another set. By studying the definition of subset we can understand that subsets are part of another set which is used a symbol `sube`. Online gives a clear definition of subsets which helps to study problems easily.

Explanation to study definition of subset:

The definition of subset is as follows.

A set X is called as a subset of a set Y if some of or all the elements of X are existing in the set Y which can be denoted as X `sube` Y. We can also write the set as the set y is a superset of set X and denoted as Y `supe` X. A empty set is also taken as a subset for any kind of set.

Representation to study definition of subset:

The elements in X are existing in the set Y called X is the subset of Y.

Example problems to study definition of subset:

Example: 1

Write the subset relation for the following sets.

A = {m, n, o} , B = {o, p} and C = {m, n, o, p, q, r}

Solution:

Given sets are,

A = {m, n, o}

B = {o, p}

C = {m, n, o, p, q, r}

A and B:

The elements in A is not in B as well as the elements in B are not in A.

A and C:

The elements in A are in C. So, A is called as subset for the set C = {m, n, o, p, q, r} which can be represented as A `sube` C.

B and C:

The elements in B are in C. So, B is called as subset for the set C = {m, n, o, p, q, r} which can be represented as B `sube` C.

Example: 2

Say whether the set P = {12, 14, 15} is a subset for a set Q = {11, 12, 13, 14, 15}.

Solution:

Given sets are,

P = {12, 14, 15}

Q = {11, 12, 13, 14, 15}

The elements in P are in Q. So, P is called as subset for the set Q = {11, 12, 13, 14, 15} which can be represented as P` sube ` Q.

My forthcoming post is on Decimals Place value and pre algebra online will give you more understanding about Algebra.

Practice problems to study definition of subset:

Problem: 1

Write the subset relation for the sets C = {9, 7} and D = {9, 8, 7, 6}

Answer: C `sube` D

Problem: 2

Write the subset relation for the sets S = { } and P = {4, 5, 10}

Answer: S `sube` P

Preparation for Subset

In mathematics subsets are the terms used in set theory. The preparation depends on the elements in the sets. For the preparation for subset we have the derive a set by having elements of another set. The subset preparation uses the symbol `sube` .

For example, C `sube` A denotes C is subset of the set A and A `supe` C denotes A is a superset of C.

Understanding subset of a set is always challenging for me but thanks to all math help websites to help me out. 

Explanation to preparation for subset:

The preparation for subset is as follows.
Every set has a subset which is derived from the set. This subset may have all elements from the given set and it may be an empty set.

For example, set V = {l, p m , n}. Some of the possible subset for the given set V are { } , {l, p, m , n} , {l}, {l , p} , { m, n} , {p, m, n} etc. In this the set V is called as superset.

Example problems to preparation for subset:

Example: 1

Prepare the subsets of a set C = {2, 3}

Solution:
Given: C = {2, 3}

Subsets has the elements from the given set C = {2, 3} such as { }, {2}, {3}, {2, 3}

Example: 2

Which of the following is true for the sets A = {1, 2, 3, 4} B = {3, 4}?

a) B `supe` A

b) A `sube` B

c) A = B

d) B` sube` A

Solution:
Given A = {1, 2, 3, 4} B = {3, 4}
        B has the elements 3, 4 which is in A and has the elements 1, 2, 3, 4 where 1, 2 are not in B.So, B is a subset of A and A is a super set of B. (B `sube` A)
Answer: d

Algebra is widely used in day to day activities watch out for my forthcoming posts on Sum of Exterior Angles Formula and Multiplication Fractions. I am sure they will be helpful.

Practice problems to preparation fo subset:

Problem: 1

Which of the following is true for the sets X = {a, c, b, m, l, i} Y = {a, b, l}?

a) Y `supe` X

b) Y `sube` X

c) Y = X

d) X `sube` Y

Answer: b
Example: 2

Which of the following is true for the sets P = {p, m} Q = {p, q, r, s}?

a) P `supe` Q

b) P `sube` Q

c) P = Q

d) Q `sube` P

Answer: b

Height of Trapezoid

Geometry deals with shapes, structures, lines, planes and angle’s. Geometry learning is also known as architectural learning. Basic shapes of geometry are square, triangle, rectangle, parallelogram, trapezoid etc. Trapezoid is one of the basic shapes in geometry. Trapezoid is a quadrilateral which has 4 sides. The total internal angle of the trapezoid is 360 degree. In trapezoid, one pair of opposite sides is parallel.

Formula for finding the height of the Trapezoid

The formula for finding the area of pyramid is given as,

Area of pyramid = h (b1 + b2)/2

Where,

h = height of the pyramid,

b1, b2 = bases of the trapezoid.

From the given area formula we can find the height of the trapezoid when the area of trapezoid is given,

A = h (b1 + b2)/2

2A = h (b1 + b2)

h = 2A/b1+ b2



Problems on height of pyramid:

Example 1:

Find the area of the trapezoid, whose bases are 10 cm and 12 cm, height, is 6 cm.

Solution:

Formula for finding the area of the trapezoid is,

Area of pyramid = h (b1 + b2)/2

= 6 (10 + 12) / 2

= 6 (22) / 2

= 3 * 22

= 66 cm2.

The answer is 66cm2.



Example 2:

Find the height of the trapezoid, whose bases are 8 cm and 12 cm, area, is 120 cm2.

Solution:

Formula for finding the area of the trapezoid is,

Area of pyramid = h (b1 + b2)/2

120 = h (8 + 12)/2

120 * 2 = h (20)

240 =   h*20

Divide 20 on both sides,

240/20 = 20*h/20

12 = h

The height is 12 cm.

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Example 3:

Find the height of the trapezoid, whose bases are 6 cm and 4 cm, area, is 100 cm2.

Solution:

Formula for finding the area of the trapezoid is,

Area of pyramid = h (b1 + b2)/2

100 = h (6 + 4)/2

100 * 2 = h (10)

200 =   h*10

Divide 10 on both sides,

200/10 = 10*h/10

20 = h

The height is 20 cm.

Least Common Multiple of 3 and 6

In mathematics, the least common multiple of two rational numbers a and b is the smallest positive rational number that is an integer multiple of both a and b. Since it is a multiple, it can be divided by a and b without a remainder. If either a or b is 0, so that there is no such positive integer, then LCM(a, b) is defined to be zero. (Source: From Wikipedia).

Least common multiple of two numbers can be found by their multiples. Here we are going to learn how to find the least common multiple of two or more numbers.

Least common multiple of 3 and 6

The least common multiple of 3 and 6 can be found by finding the multiples 3 and 6.

The list of multiples of 3 and 6 are given below

The multiples of 3 = 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60.

The multiples of 6 = 6, 12, 18, 24, 30, 36, 42, 48, 54, 60.

Here 6, 12, 18, 24, 30, 36, 42, 48, 54, 60 are the common factors, among those 6 is the lowest common number.

So, 6 is the lowest common multiple of 3 and 6.

Example problems for least common multiple

Example 1

Find the least common multiple of 3 and 16

Solution

The least common multiple of 3 and 16 can be found by finding the multiples 3 and 16.

The list of multiples of 3 and 16 are given below

The multiples of 3 = 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60.

The multiples of 16 = 16, 32, 48, 64, 80, 96, 112, 128, 144, 160.

Here 48 is the lowest common number. So, 48 is the lowest common multiple of 3 and 16.

Example 2

Find the least common multiple of 13 and 6

Solution

The least common multiple of 13 and 6 can be found by finding the multiples 13 and 6.

The list of multiples of 13 and 6 are given below

The multiples of 13 = 13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143.

The multiples of 6 = 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 72, 78, 84.

Here 78 is the lowest common number. So, 78 is the lowest common multiple of 13 and 6.

My forthcoming post is on Positive Correlation Graph and percentage formulas will give you more understanding about Algebra.

Example 3

Find the least common multiple of 13 and 16

Solution

The least common multiple of 13 and 16 can be found by finding the multiples 13 and 16.

The list of multiples of 13 and 16 are given below

The multiples of 13 = 13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143, 156, 169, 182, 195, 208.

The multiples of 16 = 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, 192, 208.

Here 208 is the lowest common number. So, 208 is the lowest common multiple of 13 and 16.

Beginner Multiplication

Multiplication (x) is the arithmetical operation of calculating one value by another value. It is a one kind of essential operations in basic arithmetic (the others operations are addition, subtraction and division). Since the outcome of calculating by whole numbers can be thinking of as including of a few number of copies of the original, whole-number products larger than 1 can be calculated by frequent addition.

Types of multiplication for beginner:

• Multiplication of variable by exponent

• Multiplication of fraction

• Multiplication of unlike signs ( positive(+),negative (-))


Understanding Multiplication Fractions is always challenging for me but thanks to all math help websites to help me out.

Multiplying Variables with Exponents:

Multiplication of exponent:

1) Exponent of 0:

If the exponent is 0 specifies you are not multiplying by anything and the answer is 1

For example, a0 = 1

x0 = 1

2) Exponent of 1:

If the exponent is 1 specifies you are multiplying the variable with 1. (Example x1 = x)

Rules for multiplication terms on fractions:

• First multiple the value of numerator.

• Then multiple the value of denominator.

• Lastly decrease the fraction (if required).

Example:

`2/5` ×` 3/4`

Step1 Multiply the numerators:

`3 / 4` × `2 / 5` = > 3 × 2 / 4 x 5 = `6/20`

Step2 Multiply the denominators:

` 3/4` ×`2/5` = 3×2 / 4 x 5 = `6/20`

Step3 Lastly decrease the fraction

Therefore solution is `3/10` .

My forthcoming post is on Population Versus Sample and The Prime Numbers will give you more understanding about Algebra.

Multiplication of unlike signs for beginner:

• Positive(+) × Positive(+) = Positive(+)

Ex:  6 × 2   =12
• Positive (+) × Negative (-) = Negative (-).

Ex:   6 × (-2) = -12
• Negative (-) × Positive (+) = Negative (-).

Ex: (-6) × 2 = -12
• Negative(-) × Negative(-) = Positive(+).

Ex: (-6) × (-2)=12

Example:

Multiply by (a + 4) (a -5)

Step 1: multiply by y in the second factor

a (a-5) =  a2- 5a

Step 2: multiply by 5 in the second factor

4(a - 5) = 4a-20

Step 3: add step 1 and step 2

(a + 4) (a - 5) = a2- 5a + 4a - 20

= a2 -1a - 20

Basic multiplication problems for beginner:

1) Multiply the values i) 23 × 2 ii) 4 × (6 × 3) = (4 × 6) ×3

Solution:

i) 23 × 2 = 46

ii)    4 × (6 × 3) = (4 × 6) ×3

4 × 18 = 24 × 3

72 = 72.

2) Multiply the values i) 4 × (2 + 6)   ii) 6 × (5 × 3) = (6 × 5)×3

Solution:

i) 4 × (2 + 6) = 4 × 2 + 4 × 6

= 8 + 24

= 32.

ii)6 × (5 × 3) = (6 × 5) × 3

6 × 15 = 30 × 3

90 = 90.

Isosceles Triangle Hypotenuse

When the two sides of the triangle are said to be equal then the triangle is called as isosceles triangle. When all the sides of the triangle are equal then the triangle is called as the equilateral triangle and when no sides of the triangle are equal then it is said to be scalene triangle. In an isosceles triangle only the sides but also the two angles are said to be equal.
                                                               
Here we will see about the isosceles triangle hypotenuse.

Please express your views of this topic Isosceles Triangle Formula by commenting on blog.

Isosceles triangle hypotenuse.

The hypotenuse side of the isosceles triangle is the sides that are found opposite to that of the right angle. According to the right angle theorem the height of the isosceles triangle is given by
                         h = √(b² – ¼ a²)
Thus the area of an isosceles triangle is
                        A = ½ ah
                            = ½ a √(b² – ¼ a²)
                            = ½ a² √((b²/a²) – (¼))
An isosceles triangle is also called as the triangle with two congruent sides. The angles which are opposite to these congruent sides are called as the base angles and the angles found between those sides are called as the vertex angle of the isosceles triangle.
Any equilateral triangle can be an isosceles triangle but no isosceles triangle is an equilateral triangle.                           

Properties of the isosceles triangle base:

• According to the right angle theorem the sum of the squares of the hypotenuse side is equal to the sum of the squares of the other two sides. The hypotenuse side is the side which is opposite to the right angle

• The side which is not equal to the other sides of a triangle is called as the base of the isosceles triangle.

• The base angles of the isosceles triangle are found to be equal.

• When the third angle of the isosceles triangle is a right angle then it is called as the right isosceles triangle.

• The perpendicular distance from the base to the vertex of an isosceles triangle is called as the altitude of the isosceles triangle.

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Externally Tangent Circles

Two intersect circles in a single point is known as tangent circles. It can be divided into two types of tangency: internal and external. By using the tangent circles many problems and constructions in geometry are solved. This type of problems have real-life applications. The followings are some real-time applications: trilateration and maximizing the use of materials.

                                             

I like to share this Tangent geometry with you all through my article. 

Illustration:

A tangent that is common to two circles and does not intersect the segment joining the centers of the circles is called Common External Tangent. A common tangent can be any one of the following:External tangent or Internal tangent.

                                                   
          From the above figure, we can see that line PQ is the common external tangent.

                                               
          From the above figure, we can see that line AB is the common external tangent.

Theorem for Externally Tangent Circles:

Theorem:If a line is tangent to a circle, it is perpendicular to the radius drawn to the point of tangency.
                                                       
Solution:
            If:        Line  AB is a tangent
                       D is point of tangent.
            Then:   OD ┴ AB

Example problems:

Example 1: Find the common external tangent from the below figure.

                                             
Solution:
            A tangent that is common to two circles and does not intersect the segment joining the centers of the circles is called common external tangent.
            From the given figure, the line CD touches both the circles on the same side of the line is the external tangent to the circle.
            Therefore, CD is the common external tangent line.

Example 2: Find the common external tangent from the below figure.
                                                                                    
Solution:
            The given figure is concentric circles. Concentric circles having the same center and not tangent.

Example 3: Find the common external tangent from the below figure.
                                                     
Solution:
          From the above figure, we can identify that given is two complete separate circles.
          The given figure have two external tangents.

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Example 4: Find the equation for tangent lines y=3x³ from the point x= 2.
Solution :
Given, y=3x³………….(1)
Step 1: To find the first derivative of y

                y ' = 3x ² ………..(2)
Step  2: To plug x =2 into y ' to find the slope at x.
                y ' = 3(2) ²
                y= 12
               Slope of tangent line =12
Step 3:To plug x=2 into y to find the y value of the tangent point
              y =3(2)³
              y=24 
              Hence tangent point (2,24)
Step 4: To plug the slope=12 and point (2,24) using the point-slope equation to find the equation     for the tangent line.
              y- y₁ =m (x-x₁)
              y- 24 =12 (x-2)
              y =12x- 24 + 24
              y=12x
Example 5: Find the slope of tangent line which is passing through the point (2, 0) and the curve y = 2x – x³
Solution:
            Step 1: Find the derivative y¹
                                y = 2x – x³
                                y¹ = 2 -3x²
            Step 2: To find the slope of tangent line,
                         Given point (2, 0) --- > (x, y)
                         y¹ = 2 -3(2)² ---- > 2 – 3(4) --- > 2 -12 = -10
                         Slope of the tangent line = -10         

Practice problems:

1. Determine the equation of the tangent line passing through the point (− 1, − 2) and having slope4 /7
Answer:   4x − 7y − 10 = 0
2. Determine the equation of the line with the slope 3 and y-intercept 4.
Answer: y = 3x + 4