Download Statements and Quantifiers (1.1).

Lesson 1.1
Getting Started…
 Let p(x) be the sentence
2x + 4 ≥ 10x – 3
 A.) Give the value of p(1)
6≥7
 B.) Find another value of x for which p(x) is not true.
x=5
 C.) Describe the set of all values of x for which p(x) is
true.
{x : x ≤ .875}
 D.) Describe the set of all values of x for which p(x) is
false.
{x: x > .875}
What is a Statement?
 A statement is a sentence that is always true or always
false, but NOT both.
 Which of these are statements?
 4x is an integer
Not a statement
2+6=7
Statement; truth value: false
 2+x=8
Not a statement
Quantifiers
 Universal Statement:
For all x in S, p(x)
 Example: For all real numbers x and y, x + y = y + x
real numbers x and y, x + y = y + x
 True / False:
A
A

A
 Symbol : For all =
integers x,
False; 0
Quantifiers (continued)
 Existential Statement: There exists x in S such that p(x).
 Example: There exists a number x st x + 5 = 8
Means “such that”
a number x st x + 5 = 8
 True/False:
E
E

E
 Symbol: There exists 
x in
Z st x
2
≥0
True
E A
Write using symbols,
, , or both
 No palm trees grow naturally in
Wisconsin.
palm trees x, x does not grow naturally in Wisconsin.
A
 Any number divided by a certain
number equals itself 
E
real numbers, r,
a number x st
A
Last Example:
 Suppose k ≥ 1. Use substitution to show that
 We know that (a + b)2 = a2 + 2ab + b2 , holds for all real
numbers a and b. Since a and b are real numbers
when k ≥ 1, this substitution works.
*Use Algebra to show!
Homework
Page 11 – 13
1-26!