#### MATH 2602

##### Midterm 1 | Spring '15| Barone
Let $A={ 1, 2, 3 }$.
(a) Give an example of a relation $R$ on $A$ which is reflexive, not symmetric, and not transitive.

(b) Explain why your relation $R$ is not transitive.

Let $f:A → B$ and $g: B → C$ be function.
(a) Prove that if $g ◦ f$ is one-to-one, then $f$ is one-to-one.

(b) Write down the converse of the statement, you proved above and give an example of function $f/g$ where it is false.

Prove that the arguments below are valid.
(a)

4.

Short answer section: put a number, statemnet, or argument in each box. Please show your work for potential partial credit.
(i) Give an example of an individual argument with at least two premises using only an atomic variable $p$.

(ii) Give an example of an implication which is always true using the atomic variables $p,q$.

(iii) Simplify the negation of the statement

(iv) Let $A={ a,b }$. Then, the number of subsets of $A$ equals

True and false questions. Instructions: for each statement below, circle TRUE if the statement is always true and circle FALSE otherwise. Your work will not be graded.
(i) The relation $\mathcal{R} = { (a, b) \in \mathbb{Z}^2 | a - b \ge 0 }$ is reflexive, anti-symmetric, and transitive. TRUE/FALSE.

(ii) The statement $q \to (\lnot p \to q)$ is neither a contradiction nor a tautology. TRUE/FALSE.

(iii) The statement $\lnot p \land (q \to r)$ is true when $p, q$, and $r$ are all true. TRUE/FALSE.

(iv) For any $a \in \mathbb{Z}$, the number $a$ is odd if and only if the number $a^2$ is odd. TRUE/FALSE.

(v) $\forall r \in \mathbb{R}( \exists n \in \mathbb{Z}(n \le r \lt n + 1))$. TRUE/FALSE.
(vi) There are $n$ equivalence classes of integers under the equivalence relation remainder modulo $n$. TRUE/FALSE.
(vii) If $A, B$ are sets and both $A \subseteq B$ and $A \cap B = \emptyset$, then $B = \emptyset$. TRUE/FALSE.