A mathematical sentence containing an equal sign is an equation. The two parts of an equation are called its members. A mathematical sentence that is either true or false but not both is called a closed sentence. To decide whether a closed sentence containing an equal sign is true or false, we check to see that both elements, and members of the sentence name the same number. To decide whether a closed sentence containing a ≠ sign is true or false, we check to see that both elements do not name the same number.
The relation of equality between two numbers satisfies the following basic axioms for the numbers a, b and c.
Reflexive: a = a.
Symmetric: If a = b then b = a.
Transitive: If a = b and b = c then a = c.
While the symbol = in an arithmetic sentence means is equal to, another symbol ≠, means is not equal to. When an = sign is replaced by ≠ sign, the opposite meaning is implied. (Thus 8 = 11 3 is read eight is equal to eleven minus three while 9 + 6 ≠ 13 is read nine plus six is not equal to thirteen.)
The important feature about a sentence involving numerals is that it is either true or false, but not both. There is nothing incorrect about writing a false sentence, in fact in some mathematical proofs it is essential that you write a false sentence.
We already know that if we draw one short line across the symbol = we change it to ≠. The symbol ≠ implies either of two things is greater than or is less than. In other words the sign ≠ in 3 + 4 ≠ 6 tells us only that numerals 3 + 4 and 6 name different numbers, but does not tell us which numeral names the greater or the lesser of the two numbers.
To know which of the two numbers is greater let us use the conventional symbol < and > . < means is less than while > means is greater than. These are inequality symbols because they indicate order of numbers. (6 < 7 is read six is less than seven, 29 > 3 is read twenty nine is greater than three). The signs which express equality or inequality (= , ≠ , < , >) are called relation symbols because they indicate how two expressions are related.
Express the symbol = , ≠ , > , < in arithmetical sentences.
e. g. x > y : Is x equal to y? No, x is greater than y.
b) α ≠ β
d) x² − x < 0
How are the symbols = , ≠ , < , > read?
− When do you use er / est
ier / iest ?
more / most
− When do you use as as
not as as
Exercise 12. Write the comparative and superlative of the words below.
Exercise 13. Write the words in brackets in the correct form of the degrees of comparison.
a. We all use this method of research because it is . (interesting) the one we followed.
b. I could solve quicker than he because the equation given to me was .. (easy) the one he was given.
c. The remainder in this operation of division is .. (great) than 1.
d. The name of Leibnitz is .. (familiar) to us as that of Newton.
e. Laptops are . (powerful) microcomputers. We can choose either of them.
f. A mainframe is (large) and .. (expensive) a microcomputer.
g. One of the (important) reasons why computers are used so widely today is that almost every big problem can be solved by solving a number of little problems.
h. Even the (sophisticated) computer, no matter how good it is, must be told what to do.
*Exercise 14. Put the words in brackets into the correct form to make an accurate description of sizes of computers.
There are different types of computer. The (large) and (powerful) are mainframe computers. Minicomputers are (small) than mainframes but are still very powerful. Microcomputers are small enough to sit on a desk. They are the (common) . type of computer. They are usually (powerful) .. than minicomputers.
Portable computers are (small) .. than desktops. The (large) portable is a laptop. (Small) portables, about the size of a piece of writing paper, are called notebook computers. Subnotebooks are (small) .. than notebooks. You can hold the (small) computers in one hand. They are called handheld computers or palmtop computers.