Example 1.1.4 Rewriting an Existential Universal Statement
Fill in the blanks to rewrite the following statement in three different ways: There is a person in my class who is at least as old as every person in my class. a. Some is at least as old as . .
b. There is a person p in my class such that p is
c. There is a person p in my class with the property that for every person q in my class, p is .
a. person in my class; every person in my class b. at least as old as every person in my class c. at least as old as q ■
Some of the most important mathematical concepts, such as the deﬁnition of limit of a sequence, can only be deﬁned using phrases that ...view middle of the document...
In each of 1–6, ﬁll in the blanks using a variable or variables to rewrite the given statement. 1. Is there a real number whose square is −1? ? a. Is there a real number x such that such that x 2 = −1? b. Does there exist 2. Is there an integer that has a remainder of 2 when it is divided by 5 and a remainder of 3 when it is divided by 6? ? a. Is there an integer n such that n has such that if n is divided by 5 the b. Does there exist ? remainder is 2 and if Note: There are integers with this property. Can you think of one?
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6 Chapter 1 Speaking Mathematically 3. Given any two real numbers, there is a real number in between. a. Given any two real numbers a and b, there is a real num. ber c such that c is b. For any two , such that a < c < b. d. If J , then J e. For all squares J , . .
4. Given any real number, there is a real number that is greater. s such that s is a. Given any real number r , there is . b. For any , such that s > r . 5. The reciprocal of any positive real number is positive. a. Given any positive real number r , the reciprocal of b. For any real number r , if r is , then . c. If a real number r , then . 6. The cube root of any negative real number is negative. a. Given any negative real number s, the cube root of b. For any real number s, if s is , then . c. If a real number s , then . .
9. For all equations E, if E is quadratic then E has at most two real solutions. . a. All quadratic equations . b. Every quadratic equation . c. If an equation is quadratic, then it , then E . d. If E . e. For all quadratic equations E, 10. Every nonzero real number has a reciprocal. . a. All nonzero real numbers for r . b. For all nonzero real numbers r , there is c. For all nonzero real numbers r , there is a real number s . such that 11. Every positive number has a...