Download (direct link):
Example 2. In how many different ways can Ann choose a team of 3 debaters from 5 volunteers if the order of the selection is taken into account? Ann may decide to make the first person, in the order of selection, the first speaker, and the second person in the order of selection the second speaker, and so on.
Solution. We proceed as before to obtain the 10 selections or combinations:
ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE
Now, however, each selection can be arranged six times; for example, one can rearrange CDE as
CDE CED DCE DEC ECD EDC
86 COMMUTATIVE LAW
The are 10 selections, and each selection can be arranged 6 times, so altogether there are 10 x 6 = 60 arrangements. There are 60 ways of choosing 3 people from 5, if the order does count. We write this permutation, or arrangement, of selecting 3 people from 5 people as 5p3, which is equal to 60.
This permutation 5p3 can be worked out using a scientific calculator to give the answer 60, without writing down all the possible arrangements.
Under this entry the meaning of three laws will be explained: the commutative law, the associative law, and the distributive law.
The commutative law involves two elements of a set, say a and b, and an operation * which can be performed on any pair of elements of the set. The commutative law states
a * b = b * a
This law emphasizes that the order of the elements can be reversed. This formal definition can be understood more easily using an example in which the set containing a and b is the set of real numbers and the operation * is multiplication.
Example 1. Is multiplication of real numbers commutative? Give two examples that verify your answer.
Solution. In these examples the commutative law involves two numbers, and the operation that combines them is multiplication. Suppose the two numbers are 5 and 7. Does 5 x 7 = 7 x 5? Yes, because 5 x 7 = 35, and 7 x 5 = 35.
Suppose the two numbers are 1.3 and 5.4. Does 1.3 x 5.4 = 5.4 x 1.3? Yes, because 1.3 x 5.4 = 7.02 and 5.4 x 1.3 = 7.02.
COMMUTATIVE LAW 87
The addition of real numbers is also commutative: for example, 3 + 6 = 6 + 3. The division of real numbers is not commutative: for example, 8 + 2 ^ 2+8, where ^ means “is not equal to.” Subtraction is also not commutative for the set of real numbers.
For an operation to be commutative for the elements of a set it must be true for all possible pairings of elements of the set. If there is one pair of elements that fails the test, then we say the set is not commutative for that operation.
The associative law involves three elements of a set, say a, b, and c, and an
operation * which can be performed on any pair of elements of the set at a time. The
associative law states
(a * b) * c = a * (b * c)
It must be remembered that the law of BEDMAS is to be applied, and the inside of the brackets is worked out first. Applying the associative law to an example will aid understanding of the concept. Suppose the set containing a, b, and c is the set of real numbers and the operation is addition. The following working suggests that the associative law is true for addition of numbers:
(3 + 5) + 2 = 3 + (5 + 2)
(3 + 5)+ 2 = 8 + 2 and 3 +(5+ 2) = 3+ 7
= 10 =10
The associative law is true for multiplication of numbers, but is not true for subtraction and division of numbers. The following example verifies that the division of numbers is not associative.
Example 2. Is this statement true?
(18+ 6)+ 2 = 18+ (6+ 2)
(18 + 6) + 2 = 3 + 2 and 18 + (6 + 2) = 18 + 3 = 1.5 =6
The distributive law involves three elements of a set, say a, b, and c, and two operations * and #. Each operation can be performed on any pair of elements of the set at a time. The distributive law states that * is distributive over # and is expressed as
a*(b#c) = a*b#a*c
An important practical use of the distributive law is in algebra and the two operations are multiplication and addition. When we expand brackets we are using the fact
88 COMPARATIVE COSTS
that multiplication is distributive over addition. Factoring is a process that uses the distributive law in reverse.
Example 3. Expand 3(x + 4).
3(x + 4) = 3 x (x + 4) Inserting the x sign
= 3xi + 3x4 Multiplication is distributive over addition.
= 3x + 12 Writing the product 3 x x without the x sign.
It is interesting to note that addition is not distributive over multiplication. This fact is demonstrated in the following example: a-\-(bxc)^a-\-bxa-\-c.
References: Addend, BEDMAS, Expanding Brackets, Operations.
COMPACT FORM OF DECIMALS
When a decimal is written in the form 0.358 it is in compact form, and when it is written as
^ X 10 +5 X 100 + ^ X 1000
it is written in expanded form. The expanded form gives meaning to the column headings in which the figures of the compact decimal are written. In a similar way, when a number is written in the form 287 it is in compact form. When it is written as 2x100+ 8x10 + 7x1 the number is in expanded form.