Real Analysis: Sets and Logic Foundations

Define an injective map and a surjective map.

Show that a map from the natural numbers to the square numbers, defined by mapping $x$ to $x^2$, is bijective.

Let $a$ be a set. We say that a family of sets is a cover of $a$ if $a$ is a subset of the union of that family of sets.

Simplify the boolean expression: $\overline{a+b}$ + $\overline{a \cdot b}$

Let $A$ be a boolean expression: $A = \overline{(A + \overline{(B \cdot C)})} \cdot \overline{(A + \overline{B})}$. Prove that $A$ simplifies to $A \cdot \overline{B}$ using DeMorgan's Theorem.

Let C be the set of all integers n such that n = 6r - 5 for some integer r. Let D be the set of all integers m such that m = 3s + 1 for some integer s. Prove or disprove: (a) C is a subset of D; (b) D is a subset of C.

Prove that set A, which consists of all integers that can be written as $4p$, is a subset of set B, which consists of all integers that can be written as $2q$, where $p$ and $q$ are integers.

Show that $(A_1 \cup A_2 \cup \cdots \cup A_n)^c = A_1^c \cap A_2^c \cap \cdots \cap A_n^c$ for any finite $n$. Use induction to prove it.

If x is an even number, then $x^2$ is an even number.

Consider the following sentence: 'This statement is false.'

Write all subsets of the set $A$, given $A = \{1, 2, 3, 4\}$.

Write all subsets of the set $B$, given $B = \{ 1, 2, \emptyset \}$.

Write all subsets of the set $C$, given $C = \{ \{\emptyset\} \}$.

Write all subsets of the empty set D.

Write all subsets of the set $E$, given $E = \{ \emptyset \}$.

Write all subsets of the set $F$, given $F = \{ \mathbb{R}, \mathbb{Q}, \mathbb{N} \}$.

Write all subsets of the set $G$, given $G = \{ \mathbb{R}, \{ \mathbb{Q}, \mathbb{N} \} \}$.

Write all subsets of the set $H$, given $H = \{ \{0, 1\}, \{0, 1, \{2\}\}, \{0\} \}$.