SOMETHING ABOUT MATHEMATICAL SENTENCES

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, or members of the sentence name the same number. To decide whether a closed sentence containing an _ 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 eigth 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 <>. <> means is greater than. These are inequality symbols because they indicate order of numbers. (6 <> 3 is read twenty nine is greater than three). The signs which express equality or inequality (=, _, <, >) are called relation symbols because they indicate how to expressions are related.