Download If x2 ≤ 0, then x ≤ 0.

Math 243
Quiz
Group work permitted
The point of these exercises is to discuss the proposition (for real numbers x) and determine whether
true or false:
If x2 ≤ 0, then x ≤ 0.
1. Common sense algebra approach:
For a real number x, suppose x2 ≤ 0. What specifically can we conclude about x?
Given this conclusion about x, is it true or false to say that x ≤ 0?
Given the comments above, is the following statement true or false, from a common sense simple
algebra point of view?
If x2 ≤ 0, then x ≤ 0.
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The point of this exercise is to discuss the proposition (for real numbers x)
If x2 ≤ 0, then x ≤ 0.
(Question 2 below is the same as Question 2 in the original worksheet. The spacing has been changed to
allow more room for your answers, and some of the puncutation may have been changed slightly.)
2. Second approach, Logic of “if-then”, by cases: Given any real number x, we there are THREE
possibilities for x,
a. Suppose x < 0. Is x2 ≤ 0 true? Is x ≤ 0 true? Explain briefly.
So, logically, is “If x2 ≤ 0, then x ≤ 0” true or false in this case? Why?
b. Suppose x = 0. Is x2 ≤ 0 true? Is x ≤ 0 true? Explain briefly.
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b. Suppose x = 0. (continued)
So, logically, is “If x2 ≤ 0, then x ≤ 0” true or false in this case? Why?
c. Suppose x > 0. Is x2 ≤ 0 true? Is x ≤ 0 true? Explain briefly.
So, logically, is “If x2 ≤ 0, then x ≤ 0” true or false in this case? Why?
d. Given the results above, would you say that “If x2 ≤ 0, then x ≤ 0” is true or false (in
general)?
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The point of this exercise is to discuss the proposition (for real numbers x)
If x2 ≤ 0, then x ≤ 0.
3. Contrapositive approach.
State the contrapositive of the proposition we are considering
If x > 0, then ... .
Is this contrapositive true or false? Explain briefly.
So, from your conclusion about the contrapositive, what can you conclude about the original
proposition,
“ If x2 ≤ 0, then x ≤ 0” ?
COMMENT: In the Homework, Exercise 2.2.2, you BEGAN with the statement “If x > 0, then ... .”
and were asked to find the contrapositive and determine the truth values. This is the “opposite” of
what has been done here.
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