9.21.2023
Question 1: The Well-Ordering Principle (WOP) states that every nonempty subset of $\mathbb{Z}^{+}$ contains ___________.
(3 points)
A. a largest element
B. a smallest element
C. a total order
D. a dual subset
Answer
B. a smallest elementQuestion 2: The induction step for the Principle of Mathematical Induction (PMI) insures which of the following logical implications (denoted by $\rightarrow$) for an arbitrary integer $k$ in $\mathbb{Z}^{+}$?
(3 points)
A. $p(k+1) \rightarrow p(k)$
B. $p(k) \rightarrow p(1)$
C. $p(k) \rightarrow p(k+1)$
D. $p(k-1) \rightarrow p(k)$
Answer
B. $p(k) \rightarrow p(1)$Question 3: Which one of the following set relations is valid for the recursively defined sets $A$ and $B$ below?
\[\{1 \in A, \; \forall y \in A, \; y + 1 \in A\}\] \[\{2 \in B, \; \forall z \in B, \; z \times 2 \in B\}\](3 points)
- $A$ is a proper subset of $B$.
- $B$ is a proper subset of $A$.
- The symmetric difference between $A$ and $B$ is the empty set.
- None of the above.