A = {x ∈ ℝ | -1 ≤ x ≤ 5}
B = {x ∈ ℤ | -1 ≤ x ≤ 5}
C = {x ∈ ℝ | x > 6}
You asked:
Given the below definitions for sets:
\[\begin{align*}& A &=& \{x \in \mathbb{R} \mid -1 \le x \le 5\} \\ & B &=& \{x \in \mathbb{Z} \mid -1 \le x \le 5\} \\ & C &=& \{x \in \mathbb{R} \mid x > 6\}\end{align*}\]Evaluate each set by simplifying it and computing its cardinality, and evaluate the union and intersection of all given sets.
MathBot Answer:
The solution to the expression $\{x \in \mathbb{R} \mid -1 \le x \le 5\}$ is the interval $\left[-1, 5\right]$.
The elements of the interval $\left[-1, 5\right]$ are not enumerable, as the interval cannot be made in one-to-one correspondence with the natural numbers.
The cardinality of the expression is uncountably infinite.
The solution to the expression $\{x \in \mathbb{Z} \mid -1 \le x \le 5\}$ is the set \[\begin{align*} &\{-1,0,1,2,3,4,5\}\end{align*}\]
The cardinality of the expression is $7$.
The solution to the expression $\{x \in \mathbb{R} \mid x > 6\}$ is the interval $\left(6, \infty\right)$.
The elements of the interval $\left(6, \infty\right)$ are not enumerable, as the interval cannot be made in one-to-one correspondence with the natural numbers.
The cardinality of the expression is uncountably infinite.
The solution to the expression $A \cup B \cup C$ is the union $\left[-1, 5\right] \cup \left(6, \infty\right)$.
The elements of the union $\left[-1, 5\right] \cup \left(6, \infty\right)$ are not enumerable, as the union cannot be made in one-to-one correspondence with the natural numbers.
The cardinality of the expression is uncountably infinite, because the cardinality of the component set $\left[-1, 5\right]$ is uncountably infinite.
The solution to the expression $A \cap B \cap C$ is $\emptyset$, the empty set.
The empty set is a set that contains no elements.
The cardinality of the expression is $0$.