Sets and Elements: Set-Builder Notation

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Questions and Answers

Given sets $A = {1, 3, 5, 7, 9}$ and $B = {2, 4, 6}$, which of the following statements accurately describes the relationship between $A$ and $B$ concerning subsets?

There exists a non-empty subset of $A$ that is also a subset of $B$.
There are no elements common between $A$ and $B$, making their intersection an empty set. (correct)
Every element in $B$ is also an element in $A$.
The intersection of $A$ and $B$ yields the universal set.

Consider the statement: 'For any natural number $n$, $n$ is even if and only if $n^2$ is even.' Which of the following accurately represents the negation of this statement using quantifiers?

$\exists n \in \mathbb{N}, n \text{ is even } \Leftrightarrow n^2 \text{ is odd}$ (correct)
$\exists n \in \mathbb{N}, n \text{ is odd } \Leftrightarrow n^2 \text{ is even}$
$\forall n \in \mathbb{N}, n \text{ is odd } \Leftrightarrow n^2 \text{ is odd}$
$\forall n \in \mathbb{N}, n \text{ is even } \Leftrightarrow n^2 \text{ is odd}$

Let $A = {5, 6, 7}$ and $B = {a, b}$. If we form the Cartesian product $A \times B$, how many elements will be in the resulting set, and what is the characteristic of these elements?

6 elements, each being an ordered pair with the first element from $A$ and the second from $B$. (correct)
8 elements, each being an unordered pair with one element from $A$ and one from $B$.
7 elements, each being a single element from either $A$ or $B$.
5 elements, each being a combination of two elements from $A$ and $B$.

Given the sets $A = {1, 3, 5, 7, 9}$, $B = {2, 4, 6}$, and $C = {1, 2, 3, 4, 5}$ within the universal set $U = {1, 2, 3, 4, 5, 6, 7, 8, 9}$, determine which expression accurately represents the elements that are exclusively in $A$ or $B$, but not in $C$.

$(A \cup B) \cap C^c$ (B) Signup and view all the answers

Consider the open interval $(-1, 7)$. Which of the following set-builder notations most accurately describes this interval?

{x \in \mathbb{R} \mid -1 < x < 7} (D) Signup and view all the answers

Given the statement 'If a function $f$ has a horizontal tangent line at $x = a$, then $f'(a) = 0$', express this statement using symbolic logic. Subsequently, identify the symbolic representation that corresponds to its contrapositive form.

$P \Rightarrow Q$; contrapositive: $\neg Q \Rightarrow \neg P$ (A) Signup and view all the answers

Consider the following sets: $A$, $B$, and $C$ within a universal set $U$. Which of the Venn diagram shading patterns accurately represents the set $(A \cap B) \cup (A \cap C) \cup (B \cap C)$?

The regions where at least two of the sets ($A$, $B$, or $C$) overlap are shaded. (D) Signup and view all the answers

You are given the statement: 'There exist real numbers $x$ and $y$, such that $x < y$ and $f(x) \geq f(y)$'. Rewrite this statement using quantifiers and logical symbols, and identify the most correct representation.

$\exists x, y \in \mathbb{R} \mid (x < y) \wedge (f(x) \geq f(y))$ (A) Signup and view all the answers

Using truth tables, demonstrate the logical equivalence of $(P \Rightarrow Q) \wedge (P \Rightarrow R)$ and $P \Rightarrow (Q \wedge R)$. Which of the following interpretations correctly evaluates this equivalence?

If $P$ implies $Q$ and $P$ implies $R$, it is logically equivalent to $P$ implying both $Q$ and $R$ occur together. (C) Signup and view all the answers

Consider the statement: 'Whenever $a$ and $b$ are real numbers with $a < b$, there is some rational number $r$ that lies between $a$ and $b$'. Rewrite this statement using quantifiers ($\forall$, $\exists$) and symbolic logic to accurately represent its meaning.

$\forall a, b \in \mathbb{R}, (a < b) \Rightarrow \exists r \in \mathbb{Q}: (a < r < b)$ (D) Signup and view all the answers

Flashcards

What does ∈ mean?

An element is a member of a set.

What does ∉ mean?

Not an element of a set.