The Venn diagram – Two-Set Problem.

In order to help solve problem involving sets, we use a diagram known as the Venn diagram. In a Venn diagram, each set is represented by an oval or circle. That is

The universal set ‘ U’ is represented by a rectangle as shown fig.1 below. The Venn diagram gives a clear picture of information. 

If  U is the universal set and A and B are two intersecting subset of the universal set then Fig.1, Fig.2, Fig.3, Fig.4, Fig.5 and Fig.6 show how to illustrate or represent the sets A and B in a Venn diagram.

Ø  Fig.1 below is A ∩ B′, i.e., elements belonging to only A.  The shaded region represents A ∩ B′.

Ø  Fig.2 is A ∩ B, i.e. elements belonging to both A and B. The shaded region represents A ∩ B.

Ø  Fig.3 is A′ ∩ B, i.e. elements belonging to only B. The shaded region represents A′ ∩ B.

Ø  Fig.4 is A U B i.e. all element in A and B. The shaded regions represent A U B.

Ø  Fig.5 is (A U B)′ i.e. elements belonging to neither A nor B. The shaded region represents (A U B) ′.

Ø  Fig.6 is (A ∩ B′) U (B ∩ A′) i.e. elements belonging to only A and only B. The shaded regions represent (A ∩ B′) U (B ∩ A′).

The mathematical expression connecting these regions is:

Where n (U) = n(A U B) + n(A U B)′.

Example 1

In a class of 36 students, 19 read Biology, 16 read Chemistry and 5 were not allowed to read both subjects. Illustrate the information on a Venn diagram. Find how many students read

1. Both Biology and Chemistry
2.  Only Chemistry
3. Only Biology
4. Only Biology and Only Chemistry

                                                       Solution

  Let U = {students in class}, this implies n (U) = 36.

Let B = {students who read Biology}, this implies n (B) = 19.

Let C = {students who read Chemistry}, this implies n(C) = 16.

Let X = {number of students who read both subjects}.

From the Venn diagram above:

1. 4 students read both subjects.
2. The number of students who  read only chemistry = 16-x

                                                                                     = 16-4

                                                                                     = 12.

3. The number of students who read only Biology = 19-x

                                                                                  = 19-4

                                                                                  = 15.

iv.   The number of students who read only Biology and only chemistry = (19-x) + (16-x)

                                                                                                                   =     15   +     12

                                                                                                                   = 27.

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4.  Operations in Modular Arithmetic (Addition, Subtraction, Multiplication and Equations.) - New!

For any further advice on your studies or any challenge on this post, please feel free to contact me at: dzaazo@gmail.com. I will appreciate it.

Negative Indices, Zero Indices and Exponents that are Fractions:

Negative Indices:

So far we have worked with positive exponents or indices. You remember from the Division Law of Indices that:

a³ ÷ a⁵ = a^{3 – 5} = a^-2                and that:

Fraction Raised to Negative Whole Exponent

Example:

Simplify the following.

Zero Indices:

Note that for any number a, except for a = 0. [thus whether a is a whole number, fraction or decimal], the rule still holds.

Complete these examples.

Simplify the following:

1.    3^{n – 1} × 3^{1 – n}

2.    3^x × 3^-x

3.    (2a²)³ × 3a­^-6

Exponents that are Fractions:

This means that you must look for the number that when multiplied by itself four times gives 16. It is written as

This is called the fourth root of 16.

Generally,

You may also like these:

1.   Indices - New!

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3.   Operations in Modular Arithmetic (Addition, Subtraction, Multiplication and Equations.) - New!

4.  The Venn diagram – Two-Set Problem. - New!

For any further advice on your studies or any challenge on this post, please feel free to contact me at: dzaazo@gmail.com. I will appreciate it.

Indices

Before students learn the concept of indices they must have the knowledge on (i) factors and (ii) multiples. At later development on indices they have to have knowledge on (iii) like terms (iv) variables (v) algebraic expression.

When a number a is multiplied by itself a number of times, the product is called a power of a. For example, a × a is the second product of a; this is written in index form as a². The number that shows the power is called the index or exponent. The index of a² is 2 and the index of a⁵ is 5. The expression a⁵ means a×a×a×a×a. In a⁶, a is called the base.

Laws of Indices:

v  1^st Law [The Law of Multiplication ]

You know that 2³ = 2 × 2 × 2 and 2⁴ = 2 × 2 × 2 × 2.

     Therefore, 2³ × 2⁴ = (2 × 2 × 2) × (2 × 2 × 2 × 2)

                                         = 2 × 2 × 2 × 2 × 2 × 2 × 2

                                         = 2⁷

The above information shows one of the basic laws of indices (multiplication law of indices). This law can be written as; 

1.           an × am = an + m

Note that in both aⁿ and a^m, the base is the same. That is ‘a’.

Now exercise under this law:

Simplify the following and write your results in an exponential form.

1.     3³ × 3⁶

2.     2⁴ × 2⁴

3.     19¹² × 19¹³

Solution

1.      You know that 3³ = 3 × 3 × 3 and 3⁶ = 3 ×3 × 3 × 3 × 3 × 3.

Therefore, 3³ × 3⁶ = 3 ×3 × 3 × 3 × 3 × 3 × 3 × 3 × 3

                            = 3⁹.

Alternatively, that is applying the law we have:

We know that aⁿ × a^m = a^n+m, where a = 3, n = 3 and m = 6.

 Then 3³ × 3⁶ = 3³⁺⁶

           = 3⁹

Now work on your own question number (2) and (3). You can submit it through the contact me.

v  2^nd Law [ The Law of Division ]

The above information shows another basic law of indices (division law of indices). This law can be written as;

Note that in both aⁿ and a^m, the base is the same. That is ‘a’.

Now follow the explanation above and simplify the following and submit it to me through the contact me:

1)    9⁸ ÷ 9⁶

2)    3¹⁰ ÷ 3² ÷ 3³

3)    (2.5)⁴ ÷ (2.5)²

v  3^rd Law [ The Law of Exponent ]

You remember that when you work with brackets,

(2³)³ = 2³ × 2³ × 2³

From the first law, you can work out:

2³ × 2³ × 2³ = 2^{3 + 3 + 3}

                    = 2⁹.

Therefore, (2³)³ = 2⁹

The above information shows another basic law of indices (division law of indices). This law can be written as;

Now follow the explanation above and simplify the following and submit it to me through the contact me:

1.     16 × (25 ÷ 64)

2.     (5³)²

Note this very carefully!

We have now learnt the three fundamental laws or rules of indices. Using a, n and m these laws are:

Ø aⁿ × a^m = a^{n + m} – The Law of Multiplication

Ø aⁿ ÷ a^m = a^{n – m} – The Law  of Division

Ø (aⁿ)^m = a^nm – Law of Exponent

There is no fast rule for addition and subtraction other than what is being studied under algebra. Thus:

a^m + aⁿ = a^m + aⁿ and a^m - aⁿ = a^m - aⁿ since they are unlike terms.

You may also like these:

1.   Modular Arithmetic - An introduction - New!

2.  Negative Indices, Zero Indices and Exponents that are Fractions: - New!

3.   Operations in Modular Arithmetic (Addition, Subtraction, Multiplication and Equations.) - New!

4.  The Venn diagram – Two-Set Problem. - New!

 


For any further advice on your studies or any challenge on this post, please feel free to contact me at: dzaazo@gmail.com. I will appreciate it.