Quiz on August 26
Discrte Mathematics I (MATH 20123)
Qiao Zhang, Department of Mathematics
Texas Christian University
0. (4 Points) Print YOUR name:
1. (2 Points) Provide a definition for increasing functions.
Solution. A (real-variable and real valued) function f (x) is called increasing if, for every two values x1 and x2 in its
domain with x1 < x2 , we always have f (x1 ) ≤ f (x2 ).
Remark. In practice, there are some ambiguities about the term “increasing”.
(1) Some authors interpret “increasing” as x1 < x2 =⇒ f (x1 ) < f (x2 ).
(2) Some authors interpret “increasing” as x1 < x2 =⇒ f (x1 ) ≤ f (x2 ).
2. (3 Points) Consider the following statement: If x > 0, then x2 > 0.
(1) Determine the converse statement.
Solution. The converse statement is: If x2 > 0, then x > 0.
(2) Determine the inverse statement.
Solution. The converse statement is: If x ≤ 0, then x2 ≤ 0.
(3) Determine the contrapositive statement.
Solution. The converse statement is: If x2 ≤ 0, then x ≤ 0.
3. (1 Point) If a statement is true, then which of the following statements must also be true?
(A) the converse statement
(B) the contrapositive statement
(C) the inverse statement
(D) the contranegative statement
Solution. B.