Problem 1: Find all subsets of $\{a,b,c\}$. Solution
The subsets of $\{a,b,c\}$ are $\varnothing,$ $\{a\},$ $\{b\},$ $\{a,b\},$ $\{c\},$ $\{a,c\},$ $\{b,c\},$ and $\{a,b,c\}.$
Problem 2: Is the empty set a subset of $X?$ Solution
Yes.
Problem 3: Is $X$ a subset of itself? Solution
Yes.
Problem 4: If $X \subseteq Y$ and $Y \subseteq Z,$ is $X$ a subset of $Z?$ In other words is this subset relation transitive? Solution
Yes.
Problem 5: Come up with two sets $S_1, S_2$ such that $S_1 \subseteq S_2,$ but $S_2 \nsubseteq S_1.$ Solution
$S_1 = \mathbb{Z},$ $S_2 = \mathbb{R}$
Problem 6: If $|X| = n,$ i.e., if $X$ has exactly $n$ elements, then how many subsets of $X$ are there? Solution
There are $2^n$ subsets of $X,$ but can you prove it?
Problem 7: If $X \subseteq Z$ and $Z \subseteq X,$ then does $X = Z?$ Solution
Yes.
Problem 8: Evaluate $\{1,2,3,4\} \cap \{1,3,5,6\}.$ Solution
$\{1,2,3,4\} \cap \{1,3,5,6\}$ $= \{1,3\}$
Problem 9: Evaluate $\{1,2,3\} \cap \{4,5,6\}.$ Solution
$\{1,2,3\} \cap \{4,5,6\}$ $= \varnothing$
Problem 10: Evaluate $[-2,5] \cap (2,7].$ Solution
$[-2,5] \cap (2,7]$ $= (2,5]$
Problem 11: Does $X \cap Z$ $= Z \cap X?$ In other words, is the operation of set intersection commutative? Solution
Yes.
Problem 12: Does $(X \cap Y) \cap Z$ $= X \cap (Y \cap Z)?$ In other words, is the operation of set intersection associative? Solution
Yes.
Problem 13: Simplify these expressions: $X \cap X,$ $X \cap U,$ $X \cap \varnothing.$ Solution
$X \cap X$ $= X,$ $X \cap U$ $= U,$ $X \cap \varnothing$ $= \varnothing$
Problem 14: Evaluate $\{1,2,3,4\} \cup \{1,3,5,6\}.$ Solution
$\{1,2,3,4\} \cup \{1,3,5,6\}$ $=\{1,2,3,4,5,6\}$
Problem 15: Evaluate $\{1,2,3\} \cup \{4,5,6\}.$ Solution
$\{1,2,3\} \cup \{4,5,6\}$ $=\{1,2,3,4,5,6\}$
Problem 16: Evaluate $[-2,5) \cup [2,7).$ Solution
$[-2,5) \cup [2,7)$ $= [-2,7)$
Problem 17: Does $X \cup Z = Z \cup X?$ In other words, is the operation of set union commutative? Solution
Yes.
Problem 18: Does $(X \cup Y) \cup Z$ $= X \cup (Y \cup Z)?$ In other words, is the operation of set union associative? Solution
Yes.
Problem 19: Simplify these expressions: $X \cup X,$ $X \cup U,$ $X \cup \varnothing.$ Solution
$X \cup X$ $= X,$ $X \cup U$ $= X,$ $X \cup \varnothing$ $= X$
Problem 20: Does $X \cap (Y \cup Z)$ $= (X \cap Y) \cup (X \cap Z)?$ In other words does intersection distribute over union? Solution
Yes.
Problem 21: Does $X \cup (Y \cap Z)$ $= (X \cup Y) \cap (X \cup Z)$? In other words does union distribute over intersection? Solution
Yes.
Problem 22: Does $X \cap (Y \cup Z)$ $= (X \cap Y) \cup Z?$ If not, give a counter example. Also, draw a venn diagram for each side of the equation. Solution
No. Counter example: $X = \varnothing,$ $Y = Z = \{1\}.$
Problem 23: Is $X \setminus U$ a subset of $X?$ Going further, is $X \setminus Z$ a subset of $X?$ Solution
Yes and yes.
Problem 24: If $X \subseteq Z$ and $Y \subseteq Z$, then is $X \cap Y$ a subset of $Z?$ Is $X \cup Y$ a subset of $Z?$ Solution
Yes and yes.
Problem 25: Is $X \cap Y$ a subset of $X?$ Solution
Yes.
Problem 26: Is $X$ a subset of $X \cup Y?$ Solution
Yes.
Problem 27: Use set builder notation to simplify $\mathbb{N} \setminus \{2n : n \in \mathbb{Z} \}.$ Solution
$\mathbb{N} \setminus \{2n : n \in \mathbb{Z} \}$ $= \{2n-1: n \in \mathbb{Z}^+\}$
Problem 28: Simplify $\mathbb{R} \setminus (0 , \infty).$ Solution
$\mathbb{R} \setminus (0 , \infty)$ $= (-\infty, 0]$
Let $\mathcal{C}$ be a collection of sets.
Problem 29: Prove $X \cup \bigcap\limits_{S \in \mathcal{C}}S$ $= \bigcap\limits_{S \in \mathcal{C}}X \cup S.$ Solution
Each side of the equation is the set of elements in $X$ or in every $S$ in $\mathcal{C}.$
Problem 30: Prove $X \cap \bigcup\limits_{S \in \mathcal{C}}S$ $= \bigcup\limits_{S \in \mathcal{C}}X \cap S.$ Solution
Each side of the equation is the set of elements in $X$ and in at least one of the $S$ in $\mathcal{C}.$
Problem 31: If $\mathcal{C} = \{(\frac{-1}{n},1 + \frac{1}{n}): n \in \mathbb{Z}^+\},$ then evaluate $\bigcap \mathcal C.$ Solution
$\bigcap \mathcal C$ $= [0,1]$
Problem 32: If $\mathcal{C} = \{[\frac{1}{n},3 - \frac{1}{n}]: n \in \mathbb{Z}^+\},$ then evaluate $\bigcup \mathcal C.$ Solution
$\bigcup \mathcal C$ $= (0,3)$
Problem 33: Prove $X \setminus \bigcap\limits_{S \in \mathcal{C}}S$ $= \bigcup\limits_{S \in \mathcal{C}}(X \setminus S).$ Solution
Each side of the equation is the set of elements in $X$ but not in every $S$ in $\mathcal{C}.$
Problem 34: Prove $X \setminus \bigcup\limits_{S \in \mathcal{C}}S$ $= \bigcap\limits_{S \in \mathcal{C}}(X \setminus S).$ Solution
Each side of the equation is the set of elements in $X$ but not in any $S$ in $\mathcal{C}.$
+10 Points