lec0905 - The Use of Quantifiers Open Statement: One that...

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The Use of Quantifiers Open Statement: One that contains a variable, and becomes a (single) statement when that variable is replaced with a value. Example of an open statement: 7 divides evenly into x+7. As you can see, for certain values of x, such as 7 or 21, this statement is true, but for other values, it is not true. (Note: It is possible for an open statement to always be true, such as “x is greater than x-1.”) We can denote the open statement above as p(x). Thus we can say that p(7) is true, whereas p(3) is not. It is also possible for open statements to contain more than one variable. Consider the following: x is a prime number that divides into y evenly AND is less than or equal to y. We can denote the statement above as q(x,y). So, q(2,18) is true while q(5,13) and q(27,3) are both false. With both of these open statements, we see that there are values for which the statements are true. Thus, it is reasonable to say something like the following: For some x, p(x), and for some x and y, q(x,y).
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Quantifier Notation Similarly, certain open statements can be true for all values of the variables involved. If we let r(x) be the statement: “x is greater than x-1,” then we can make the claim: For all x, r(x). (Of course, whenever we say for all x, we must define what our universe, or possible x’s are. Are they all the integers? All the real numbers? All the complex numbers? We will talk about how to specify that later.) Since we are dealing with mathematics, we need short symbols to replace the English words “for some” and “for all”. (A mathematicians whole goal in life is to confuse non- mathematicians :)) The symbol for the words “for some”, which is more
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lec0905 - The Use of Quantifiers Open Statement: One that...

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