In Class: Sets

Inverse, Converse, Contrapositive

Let $A$ and $B$ be sets. Define the following two propositions:

$p$: $a \in A$.
$q$: $a \in A \cap B$.

Consider the logical expression $p \rightarrow q$. We have the:

Inverse: $\lnot p \rightarrow \lnot q$.
Converse: $q \rightarrow p$.
Contrapositive: $\lnot q \rightarrow \lnot p$.

Write each of these out, replacing $p$ and $q$ with their definitions in terms of set memberships.

Determine which of the original, inverse, converse, and contrapositive statements are true using Venn diagrams
Determine which of these statements are true using what we know about logical equivalences for inverse, converse, and contrapositive. You’ll still need to know the truth value of the first statement – you can do that one with Venn diagrams.

Recall that $\mathbb{Z} = \{\ldots,-2, -1, 0, 1, 2 \ldots\}$ (the integers). Let $\mathbb{Z}^{+} = \{1, 2, 3 \ldots\}$ be the set of positive integers. Let $2\mathbb{Z}$ be the set of even integers, $3\mathbb{Z}$ the set of integers that are multiples of 3, and so on.

Is $\mathbb{Z}^+ \subseteq \mathbb{Z}$? Why or why not?
Is $2\mathbb{Z} \subseteq \mathbb{Z}^+$? Why or why not?
Find $2\mathbb{Z} \cap 3\mathbb{Z}$. Describe this set in words, using set notation, and using the notation defined in this problem.

DMOI 0.3.13

Recall that $\mathbb{Z} = \{\ldots,-2, -1, 0, 1, 2 \ldots\}$ (the integers). Let $\mathbb{Z}^{+} = \{1, 2, 3 \ldots\}$ be the set of positive integers. Let $2\mathbb{Z}$ be the set of even integers, $3\mathbb{Z}$ the set of integers that are multiples of 3, and so on.

Describe the set $\{1, 4, 7, 10, \ldots\}$ in set-builder notation. Do this two ways: once using the notation $k\mathbb{Z}$ defined above, and once by not using this notation.
Recall that, for integers $x$, $y$, and $zY, $x \equiv y \pmod z$ if $x$ and $y$ have the same integer remainder when divided by $z$. Describe the set $\{x \in \mathbb{Z} : x \equiv 3 \pmod 4\}$ in set-builder notation without using the $\pmod{}$ operator.