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CLOSURE PROPERTY
1. If we add two natural numbers, the sum will also be a natural number. For example, 5 is a natural
number and 3 is a natural number. The sum of these two numbers, 8, is also a natural number. Following
are other examples in which two natural numbers are being added and the sum is another natural number.
19 + 4 = 23 and only 23; 6 + 6 = 12 and only 12. In fact, if you add any two natural numbers, the sum is
again a natural number. Because this is true, we say that the set of natural numbers is closed under
addition.
2. Notice that in each the above equation we were able to name the sum. That is, the sum of 5 and 3
exists or there is a number which is the sum of 19 and 4. In fact, the sum of any two numbers exists. This
is called the existence property.
3. Notice also that if you are to add 5 and 3, you will get 8 and only 8 and not some other number.
Since there is one and only one sum for 19 + 4, we say that the sum is unique. This is called the
uniqueness property.
4. Both uniqueness and existence are implied in the definition of closure.
5. Now, let us state the closure property of addition.
6. If a and b are numbers of a given set, then a + b is also a number of that same set. For example,
if a and b are any two natural numbers, then a + b exists, it is unique, and it is again a natural number.
7. If we use the operation of subtraction instead of the operation of addition, we shall not be able to
make the statement we made above. If we are to subtract natural numbers, the result is sometimes a
natural number, and sometimes not. 11 – 6 = 5 and 5 is a natural number, while 9 – 9 = 0 and 0 is not a
natural number.
8. Consider the equation 4 – 7 = n. We shall not be able to solve it if we must have a natural
number as an answer. Therefore, the set of natural numbers is not closed under subtraction.
9. What about the operation of multiplication? Find the product of several pairs of natural numbers.
Given two natural numbers, is there always a natural number which is the product of the two numbers?
10. Every pair of natural numbers has a unique product which is again a natural number. Thus the
set of natural numbers is closed under multiplication.
11. In general, the closure property may be defined as follows: if x and y are any elements, not
necessarily the same, of set A (A capital) and * (asterisk) denotes an operation *, then set A is closed
under the operation asterisk if (x*y) is an element of set A.
12. To summarize we shall say that there are two operations, addition and multiplication, for which
the set of natural numbers is closed. Given any two natural numbers x and y, x + y and x × y are again
natural numbers. This implies that the sum and the product of two natural numbers exist. It so happens
that with the set of natural numbers (but not with every mathematical system) the results of the operations
of addition and multiplication are unique.
13. It should be pointed out that it is practically impossible to find the sum or the product of every
possible pair of natural numbers. Hence, we have to accept the closure property without proof, that is, as
an axiom.
Ex. 1 Определите соответствуют ли данные предложения тексту. (True / False)
1. If you add any two natural numbers, the sum is a rational number.
2. The set of natural numbers is closed under addition and multiplication.
3. It is practically possible to find the sum or the product of every possible pair of natural numbers.
4. With the set of natural numbers the results of the operations of addition and multiplication are unique.
Ex. 2 Выберите правильный вариант ответа на следующие вопросы в соответствии с
текстом.
1. What properties are implied in the definition of closure?
a) uniqueness
b) existence
c) both
2. What product does every pair of natural numbers have?
a) natural
b) unique
c) rational
3. Is it possible to find the sum or the product of every possible pair of natural number?
a) Yes, it is.
b) No, it isn’t.
c) Yes, but with some exceptions.
Ex. 3 Выберите правильный вариант перевода.
1. If you add any two natural numbers, the sum is again a natural number.
a) Если вы добавите два натуральных числа, сумма – снова натуральное число.
b) Если сложить два любых натуральных числа, сумма тоже будет натуральным числом.
c) При условии, что складываются два натуральных числа, сумма тоже натуральное число.
2. Notice that in each the above equation we were able to name the sum.
a) Заметьте, что в каждом из вышеприведенных равенств мы смогли назвать сумму.
b) Примите во внимание, что в каждом уравнении наверху мы были в состоянии назвать сумму.
c) Заметки в каждом из вышеприведенных равенств смогли назвать сумму.
3. We shall not be able to solve it if we must have a natural number as an answer.
a) Мы не сможем решить его, если в ответе должно получиться натуральное число.
b) Мы будем не в состоянии решить это, при условии, что мы должны иметь натуральное число в
качестве ответа.
c) Мы не будем решать это, так как в качестве ответа должно получиться натуральное число.
Ex. 4 Вставьте в предложения слова в соответствии с текстом.
1. Actually, the … of any two numbers exists.
а) quotient
b) difference
c) sum
2. We have to accept the closure property without … .
a) proof
b) theorem
c) axiom
3. In fact, the sum of any two numbers exists. This is called the … .
a) existence property
b) non-existence property
c) closure property
4. … of two natural numbers exist.
a) The sum and the difference
b) The sum and the product
c) The product and the difference
5. If we use the operation of subtraction instead of the operation of addition, we shall not be able to make
the … we made above.
a) sentence
b) statement
c) proclamation