# How to Prove It - Solutions

## Chapter - 3, Proofs

### Summary

• To prove a goal of the form $P ∧ Q$:
Prove $P$ and $Q$ separately.
• To use a given of the form $P ∧ Q$:
Treat this given as two separate givens: $P$, and $Q$.
• To prove a goal of the form $P ↔ Q$ :
Prove $P → Q$ and $Q → P$ separately.
• To use a given of the form $P ↔ Q$ :
Treat this as two separate givens: $P → Q$, and $Q → P$.
• Sometimes in a proof of a goal of the form $P ↔ Q$ the steps in the proof of $Q → P$ are the same as the steps used to prove $P → Q$, but in reverse order. These two directions can be summed up by proving equivalences(biconditional) in the steps involved. For eg: If such proof involves R, then it can be summed up as: P ↔ R ↔ Q.
• To prove equivalences of sets, say $A = B$, most of the proofs involve either of the following approaches:
• $\forall x (x \in A ↔ x \in B)$.
• $(A \subseteq B) \land (B \subseteq A)$.

Soln1

($\to$) Suppose $y$ is arbitrary. Since $\forall x (P(x) \land Q(x))$, it follows that $P(y) \land Q(y)$. Now since $P(y)$, and $y$ is arbitrary, it follows that $\forall x P(x)$. Similarly since $Q(y)$, and $y$ is arbitrary, it follows that $\forall x Q(x)$. Thus we can conclude that $\forall x P(x) \land \forall x Q(x)$.

$( \leftarrow$) Suppose $y$ is arbitrary. Since $\forall x P(x)$, it follows that $P(y)$. Similarly, since $\forall x Q(x)$, it follows that $Q(y)$. Thus we have $P(y) \land Q(y)$. Now as $y$ is arbitrary, we can conclude that $\forall x (P(x) \land Q(x))$.

Soln2

Suppose $x \in A$. Since $A \subseteq B$, it follows that $x \in B$. Also since $A \subseteq C$, it follows that $x \in C$. Now since $x \in B$ and $x \in C$, it follows that $x \in A \to x \in B \cap C$. As $x$ is arbitrary, we can conclude that $A \subseteq (B \cap C)$.

Soln3

Suppose $x \in C \setminus B$. Thus we have $x \in C \land x \notin B$. Since $A \subseteq B$ and $x \notin B$, it follows that $x \notin A$. Thus we have $x \in C \land x \notin A$. Thus we can say $x \in C \setminus B \to x \in C \setminus A$. As $x$ is arbitrary, we can conclude that $C \setminus B \subseteq C \setminus A$.

Soln4

Given that $A \subseteq B$ and $A \nsubseteq C$. Since $A \nsubseteq C$, suppose $x$ is an arbitrary variable such that $x \in A \land x \notin C$. Since $A \subseteq B$, it follows that $x \in B$. Thus $x \in B \land x \notin C$. As $x$ is arbitrary, we can conclude that $B \nsubseteq C$.

Soln5

Given that $A \subseteq B \setminus C$. Suppose $x \in A$. Then $x \in B \land x \notin C$. Thus there exists an element such that $x \in B \land x \notin C$, so we can conclude that $B \nsubseteq C$.

Soln6

Suppose $x \in (A \setminus (B \cap C))$. Thus we have:
$\quad = x \in A \land \lnot (x \in (B \cap C))$
$\quad = x \in A \land \lnot (x \in B \land x \in C)$
$\quad = x \in A \land (x \notin B \lor x \notin C)$
$\quad = (x \in A \land x \notin B) \lor (x \in A \land x \notin C)$
$\quad = x \in (A \setminus B) \lor x \in (A \setminus C)$
$\quad = x \in (A \setminus B) \cup (A \setminus C))$

Since $x$ is arbitrary, we can say that $A \setminus (B \cap C) = (A \setminus B) \cup (x \in A \setminus C)$.

Soln7

($\to$)Suppose $x \in \mathcal P(A \cap B)$. Then we have $x \subseteq (A \cap B)$. Suppose $y \in x$, then $y \in (A \cap B)$. Or we can say that $y \in A$ and $y \in B$. Since $y$ is arbitrary, $x \subseteq A \land x \subseteq B$.
Since $x \subseteq A$, then $x \in \mathcal P(A)$. Similarly, since $x \subseteq B$, then $x \in \mathcal P(B)$
Thus we can conclude that $x \in (\mathcal P (A) \cap \mathcal P(B))$.

($\leftarrow$)Suppose $x \in \mathcal P (A) \cap \mathcal P(B)$. Since $x \in \mathcal P (A)$, it follows that $x \subseteq A$. Similarly it can be seen that $x \subseteq B$. Now suppose $y \in x$. Then we have $y \in A$ and $y \in B$. Or we can say $y \in A \cap B$. As $y$ is arbitrary, we can say that $x \subseteq A \cap B$. Thus we can conclude $x \in \mathcal P(A \cap B)$.

Soln8

($\to$)Suppose $x \in \mathcal P(A)$, then $x \subseteq A$. Since $A \subseteq B$, it follows that $x \subseteq B$. Now since $x \subseteq B$, then $x \in \mathcal P(B)$. Thus if $x \in \mathcal P(A)$ then $x \in \mathcal P(B)$. We can conclude that $\mathcal P(A) \subseteq \mathcal P(B)$.

($\leftarrow$ )Suppose $x \in A$. Suppose $y$ is a set in $\mathcal P(A)$ such that $x \in y$. Since $\mathcal P(A) \subseteq \mathcal P(B)$, it follows that $y \in \mathcal P(B)$. Thus $y \subseteq B$. Now since $x \in y$, it follows that $x \in B$. Thus we can conclude $x \in A \to x \in B$. As $x$ is arbitrary, thus we have $A \subseteq B$.

Soln9

Suppose $x = 2i + 1$ and $y = 2j + 1$ where $i,\,j$ are integers. Then $xy = (2i+1)(2j+1) = 4ij + 2i + 2j + 1 = 2(2ij + i + j) + 1$. Suppose $k = (2ij + i + j)$, then $k$ is an integer. Thus $xy = 2k + 1$, is odd.

Soln10

($\to$)Suppose $n$ is even. Then $n = 2i$, where $i \in \mathbb Z$. Thus $n^3 = {(2i)}^3 = 8i^3$. Suppose $j = 4i^3$, then $n^3 = 2j$. Since $j \in \mathbb Z$, it follows that $n^3$ is even.

($\leftarrow$ )As an integer can either be even or odd but not both, we shall prove this by contrapositive. Thus we shall prove that If $n$ is not even(odd) then $n^3$ is also not even(odd). Suppose $n$ is odd, then $n = 2i + 1$. Thus $n^3 = {(2i + 1)}^3 = 8i^3 + 12i^2 + 6i + 1 = 2(4i^3 + 6i^2 + 3i) + 1$. Suppose $j = 4i^3 + 6i^2 + 3i$, then $j \in \mathbb Z$. Thus $n^3 = 2j + 1$ is odd. Thus we can conclude that if $n$ is even then $n^3$ is also even.

Soln11

(a)

The problem with the proof is with the usage of variable $k$ in $m = 2k$ and $n = 2k + 1$. Value of $k$ may not be same.

(b)

For $m = 0$ and $n = 3$, then $n^2 - m^2 = 9$ while $m + n = 3$. Thus $n^2 - m^2 \neq m + n$.

Soln12

($\to$) Suppose $x$ is arbitrary. Suppose $y = \frac x {x-1}$, then $x + y = x + \frac x {x-1}$
$\quad = \frac {x(x-1) + x} {x-1}$
$\quad = \frac {x^2} {x-1}$
$\quad = x \times \frac x {x-1}$
$\quad = xy$

Thus the equation: $\exists y (x + y = xy)$ is correct for $y = \frac x {x-1}$. Now since $y = \frac x {x-1}$, it follows that $x \neq 1$.

($\leftarrow$ ). We shall prove contra-positive. Suppose $x = 1$. Now $x + y = 1 + y$ and $xy = y$. Since $1 + y \neq y$, it follows that $\exists y (x + y = xy)$ is not correct. Thus if $x + y = xy$, then $x \neq 1$.

Soln13

($\to$)Suppose $z = 1$. Suppose $x \in {\mathbb R}^+$. Suppose $\exists y (y - x = \frac y x)$, then $y = \frac {x^2} {x-1}$. Now $y = \frac {x^2} {x-1}$ is defined only if $x \neq 1$. Thus there exists a $z$ such that $x \neq z$. Now since $x$ is arbitrary, it follows that $\exists z \forall x \in {\mathbb R}^+[\exists y (y - x = \frac y x) \to x \neq z ]$.

($\leftarrow$ ). Suppose $z = 1$. Suppose $x \in {\mathbb R}^+$ and $x \neq 1$. Since $x \neq 1$, we can assume $y = \frac {x^2} {x-1}$. Then $y - x = \frac y x$ is correct(can be verified by putting the value of $y$). Thus there exists a $z = 1$ for which $x \neq z$ and $\exists y (y - x = \frac y x)$ is correct. We can conclude $\exists z \forall x \in {\mathbb R}^+[x \neq z \to \exists y (y - x = \frac y x)]$.

Soln14

Suppose $x \in \cup \{A \setminus B \;\vert\;A \in \mathcal F \}$. Thus there exists a set, say $A$, in $\mathcal F$ such that $x \in A$. Since $x \in A \land x \notin B$, it follows that $A \nsubseteq B$. Since $A \nsubseteq B$, then $A \notin \mathcal P(B)$. Thus we have $A \in \mathcal F \land A \notin \mathcal P(B)$. Since $x \in A$, it follows that $x \in \cup(\mathcal F \setminus \mathcal P(B))$. Since $x$ is arbitrary, we can conclude $\cup \{A \setminus B \vert\;A \in F \} \subseteq \cup(\mathcal F \setminus \mathcal P(B))$.

Soln15

Suppose $\cup \mathcal F$ and $\cap \mathcal G$ are not disjoint. Then there must be an element, say $x$, such that $x \in \cup \mathcal F$ and $x \in \cap \mathcal G$. Since $x \in \cup \mathcal F$, then there must exist a set, say $A$, in $\mathcal F$ such that $x \in A$. Also since $x \in \cap \mathcal G$, it follows that $x$ must exist in all the sets of $\mathcal G$. Thus there exist a set $A$ such that it is not disjoint with any of the sets of $\mathcal G$. But this contradicts the given that there is no such set. Thus we can conclude that $\cup \mathcal F$ and $\cap \mathcal G$ are disjoint.

Soln16

($\to$)Suppose $x \in A$. Then $x$ exists in atleast one of the subsets of $A$. Thus $x$ must exists in $\cup\{$ all subsets of $A \}$. It follows that $x \in \cup \mathcal P(A)$. Since $x$ is arbitrary, we can conclude that $A \subseteq \cup \mathcal P(A)$.

($\leftarrow$ ). Suppose $x \in \cup \mathcal P(A)$. Thus $x$ exists in atleast one of the subsets of $A$. It follows that $x \in A$. Since $x$ is arbitrary, we can conclude that $\cup \mathcal P(A) \subseteq A$.

Thus we can say that $A = P(A)$.

Soln17

(a)

Suppose that $x \in \cup(\mathcal F \cap \mathcal G)$. Thus there exist a set, say $A$, such that $x \in A$ and $A \in \mathcal F$ and $A \in \mathcal G$. Since $A \in \mathcal F$, it follows that $x \in \cup \mathcal F$. Similarly since $A \in \mathcal G$, it follows that $x \in \cup \mathcal G$. Thus $x \in \cup F \land x \in \cup G$. It follows that $x \in (\cup \mathcal F \cap \cup \mathcal G)$. Since $x$ is arbitrary, we can conclude that $\cup(\mathcal F \cap \mathcal G) \subseteq (\cup \mathcal F \cap \cup \mathcal G)$.

(b)

The issue is that set $A$ may not be same in $\mathcal F$ and $\mathcal G$.

(c)

$\mathcal F = \{ A_1, A_2 \}$ where $A_1 = \{ 1 \}$ and $A_2 = \{ 3, 4 \}$

$\mathcal G = \{ B_1, B_2, A_2 \}$ where $B_1 = \{ 2 \}$, $B_2 = \{ 1 , 2 \}$ and $A_2 = \{ 3, 4 \}$

$\cup(\mathcal F \cap \mathcal G) = \{ 3, 4 \}$ .

$\cup \mathcal F \cap \cup \mathcal G = \{1, 3, 4 \} \cap \{1, 2, 3, 4 \} = \{1, 2, 3, 4 \}$.

Soln18

($\to$)Suppose $\forall A \in \mathcal F\;\forall B \in \mathcal G (A \cap B) \subseteq \cup (\mathcal F \cap \mathcal G)$. Suppose $x \in (\cup \mathcal F) \cap (\cup \mathcal G)$. Thus there exist some set, say $P \in \mathcal F$ such that $x \in P$. Similarly there exist some set, say $Q \in \mathcal G$, such that $x \in Q$. Thus $x \in P \cap Q$. As $P \in F$ and $Q \in G$, then $P \cap Q \subseteq \cup (\mathcal F \cap \mathcal G)$. It follows that if $x \in (\cup \mathcal F) \cap (\cup \mathcal G)$, then $x \in \cup (\mathcal F \cap \mathcal G)$. Since $x$ is arbitrary, $(\cup \mathcal F) \cap (\cup \mathcal G) \subseteq \cup (\mathcal F \cap \mathcal G)$. Thus we can conclude that: $\forall A \in \mathcal F\;\forall B \in \mathcal G (A \cap B) \subseteq \cup (\mathcal F \cap \mathcal G) \quad \to \quad (\cup \mathcal F) \cap (\cup \mathcal G) \subseteq \cup (\mathcal F \cap \mathcal G)$.

($\leftarrow$) Suppose $(\cup \mathcal F) \cap (\cup \mathcal G) \subseteq \cup (\mathcal F \cap \mathcal G)$. Suppose that $x \in A \cap B$ where $A \in \mathcal F$ and $B \in \mathcal G$. Since $x \in A$ and $A \in \mathcal F$, it follows that $x \in \cup \mathcal F$. Similarly, since $x \in B$ and $B \in \mathcal G$, it follows that $x \in \cup \mathcal G$. Thus $x \in (\cup \mathcal F) \cap (\cup \mathcal G)$. Since $(\cup \mathcal F) \cap (\cup \mathcal G) \subseteq \cup (\mathcal F \cap \mathcal G)$, it follows that $x \in \cup (\mathcal F \cap \mathcal G)$. Since $x$ is arbitrary, it follows that $\forall A \in \mathcal F\;\forall B \in \mathcal G (A \cap B) \subseteq \cup (\mathcal F \cap \mathcal G)$. Thus we can conclude that $(\cup \mathcal F) \cap (\cup \mathcal G) \subseteq \cup (\mathcal F \cap \mathcal G) \quad \to \quad \forall A \in \mathcal F\;\forall B \in \mathcal G (A \cap B) \subseteq \cup (\mathcal F \cap \mathcal G)$.

Soln19

($\to$)Suppose $A$ and $B$ are not disjoint. Then there must exist an element, say $x$, such that $x \in A$ and $x \in B$. Since $A \in \mathcal F$, it follows that $x in \cup \mathcal F$. Similarly since $B \in \mathcal G$, it follows that $x in \cup \mathcal G$. Thus $x \in (\cup \mathcal F) \cap (\cup \mathcal G )$. But it is a contradiction to the given that $(\cup \mathcal F) \cap (\cup \mathcal G ) = \phi$. Thus $\forall A \in \mathcal F\; \forall B \in \mathcal G (A \cap B) = \phi$.

($\leftarrow$)Suppose $\cup \mathcal F$ and $\cup \mathcal G$ are not disjoint. Then there must exist an element, say $x$, such that $x \in \cup \mathcal F$ and $x \in \cup \mathcal G$. Thus there must exist a set, say $A$ in $\mathcal F$ such that $x \in A$. Similarly there must exist a set, say $B$, in $\mathcal G$ such that $x \in B$. Thus $x \in A \cap B$. This contradicts to the given that $A \cap B = \phi$. Thus $\cup \mathcal F$ and $\cup \mathcal G = \phi$.

Soln20

(a)

Suppose $x \in (\cup \mathcal F) \setminus (\cup \mathcal G)$. Thus $x$ is in $(\cup \mathcal F)$ and $x$ is not in $(\cup \mathcal G)$. Since $x \in (\cup \mathcal F)$, there must exist a set, say $A$, in $\mathcal F$ such that $x \in A$. Also since $x$ not in $(\cup \mathcal G)$, there is no set, say $B$, in $\mathcal G$, such that $x \in B$. It follows that $x$ can exist in only those sets in $\mathcal F$ which are not present in $G$. Or we can say that $x$ exists in atleast one of the sets in $\mathcal F \setminus \mathcal G$. It follows that $x \in \cup (\mathcal F \setminus \mathcal G)$. Since $x$ is arbitrary, we can conclude that $(\cup \mathcal F) \setminus (\cup \mathcal G) \subseteq \cup (\mathcal F \setminus \mathcal G)$.

(b)

Statement $x ∈ A$ and $A \notin G$, then $x \notin \cup G$ is wrong. It is possible that $x$ belongs to some other set($\neq A$) of $\mathcal G$.

(c)

Lets first prove direction $\to$:

$\forall A \in (\mathcal F \setminus \mathcal G)\, \forall B \in \mathcal G ((A \cap B) = \phi) \; \to \; \cup (\mathcal F \setminus \mathcal G)\subseteq (\cup \mathcal F) \setminus (\cup \mathcal G)$.

Suppose $\forall A \in (\mathcal F \setminus \mathcal G)\, \forall B \in \mathcal G (A \cap B) = \phi$. Suppose $x \in \cup (\mathcal F \setminus \mathcal G)$. Thus there exist a set, say $P$, such that $P \in \mathcal F$ and $P \notin \mathcal G$ , and $x \in P$. As $P \notin \mathcal G$, there may exist some other set, say $Q$, such that $Q \in \mathcal G$, and $x \in Q$. Thus if such set $Q$ exists, then $P \cap Q \neq \phi$, because $x$ exists in both $P$ and $Q$. But since $\forall A \in (\mathcal F \setminus \mathcal G)\, \forall B \in \mathcal G (A \cap B) = \phi$ it follows that $P \cap Q = \phi$. Thus there is no such set $Q$. It follows that $x$ does not exist in any of the sets of $\mathcal G$. Or we can say that $x \notin \cup \mathcal G$. Since $x \in P$, it follows that $x \in \cup \mathcal F$. Thus it follows that $x \in \cup \mathcal F \land x \notin \cup \mathcal G$, or we can say that $x \in (\cup \mathcal F) \setminus (\cup \mathcal G)$.

Now proof for direction $\leftarrow$:

$\cup (\mathcal F \setminus \mathcal G)\subseteq (\cup \mathcal F) \setminus (\cup \mathcal G) \; \to \; \forall A \in (\mathcal F \setminus \mathcal G)\, \forall B \in \mathcal G (A \cap B = \phi)$.

Here we shall prove by contra-positive. Suppose that $\forall A \in (\mathcal F \setminus \mathcal G)\, \forall B \in \mathcal G ((A \cap B) \neq \phi)$. Suppose that $x \in A \cap B$, where $A \in (\mathcal F \setminus \mathcal G)$ and $B \in \mathcal G$. Thus there exist a set($A$) in $\mathcal F \setminus \mathcal G$ such that $x$ belongs to that set. It follows that $x \in \cup (\mathcal F \setminus \mathcal G)$. Also, since $x \in B$ and $B \in \mathcal G$, it follows that $x \in \cup \mathcal G$. Since $x \in \cup \mathcal G$, it follows that $x \notin (\cup \mathcal F) \setminus (\cup \mathcal G)$. Thus $\cup (\mathcal F \setminus \mathcal G)\nsubseteq (\cup \mathcal F) \setminus (\cup \mathcal G)$.Thus by contrapositive we can conclude that if $(\cup \mathcal F) \setminus (\cup \mathcal G) \subseteq (\cup \mathcal F) \setminus (\cup \mathcal G)$, then $\forall A \in (\mathcal F \setminus \mathcal G)\, \forall B \in \mathcal G (A \cap B = \phi$.

(d)

$\mathcal F = \{ A_1, A_2 \}$, where $A_1 = \{ 1 \}$ and $A_2 = \{ 10, 20 \}$.

$\mathcal G = \{ B_1, B_2 \}$, where $B_1 = \{ 1, 3 \}$ and $B_2 = \{ 10, 30 \}$.

$\cup (\mathcal F \setminus \mathcal G) = \{ 1, 10, 20 \}$.

$(\cup \mathcal F) \setminus (\cup \mathcal G) = \{ 20 \}$.

Soln21

Suppose $\cup \mathcal F \nsubseteq \cup \mathcal G$. Suppose $x \in \cup \mathcal F$, then $x \notin \cup \mathcal G$. Since $x \in \cup \mathcal F$, it follows that there exist a set, say $P$, in $\mathcal F$ such that $x \in P$. Also since $x \notin \cup \mathcal G$, it follows that $x$ does not belong to any of the sets of $\mathcal G$. Thus $\exists P \in \mathcal F \, \forall Q \in \mathcal G\,(P \nsubseteq Q)$.

Soln22

(a)

• Strategy to prove a goal of the form $P \leftrightarrow Q$.
• Strategy to prove $P \to Q$.
• Strategy to prove $P \land Q$.
• Strategy for $\forall x P(x)$.

(b)

Suppose $x \in B \setminus (\cup_{i \in I} A_i)$
It is equivalent to:
$\quad = x \in B \land \lnot \exists i \in I (x \in A_i)$
$\quad = x \in B \land \forall i \in I (x \notin A_i)$
(Note that $\, I \ne \phi \,$)
$\quad = \forall i \in I (x \in B \land x \notin A_i)$
$\quad = \forall i \in I (x \in B \setminus A_i)$
$\quad = x \in \cap_{i \in I}(B \setminus A_i)$

Since $x$ is arbitrary, it follows that $B \setminus (\cup_{i \in I} A_i) = \cup_{i \in I}(B \setminus A_i)$.

(c)

Suppose $x \in B \setminus (\cap_{i \in I} A_i )$
It is equivalent to:
$\quad = x \in B \land \lnot \forall i \in I ( x \in A_i)$
$\quad = x \in B \land \exists i \in I (x \notin A_i)$
$\quad = \exists i \in I (x \in B \land x \notin A_i)$
$\quad = \exists i \in I (x \in B \setminus A_i)$
$\quad = x \in \cup_{i \in I}(B \setminus A_i)$

Since $x$ is arbitrary, it follows that $B \setminus (\cap_{i \in I} A_i ) = \cup_{i \in I}(B \setminus A_i)$.

Soln23

(a)

Suppose $x \in \cup_{i \in I}(A_i \setminus B_i)$. Thus there must exist a set, say $A_k$, in $\{ A_i \, \vert \, i \in I \}$ such that $x \in A_k$. Thus $x \in \cup_{i \in I} A_i$. Now suppose $x$ exist in some set $B_l$ where $i = l$. Now if $A_l$ also contain $x$ then $x$, which is clearly not possible as then $x$ can not exist in $A_l \setminus B_l$. Else if $A_l$ does not contain $x$, then clearly $x$ can not exist in $A_l \setminus B_l$. Thus $x$ does not exist in any set $B_i$. It follows that $x \notin \cap_{i \in I} B_i$. Thus we can conclude that $x \in \cup_{i \in I} A_i \land x \notin \cap_{i \in I} B_i$, or $x \in \cup_{i \in I} A_i \setminus \cap_{i \in I} B_i$. Since $x$ is arbitrary, we can conclude that $\cup_{i \in I}(A_i \setminus B_i) \; \subseteq \; \cup_{i \in I} A_i \setminus \cap_{i \in I} B_i$.

(b)

$A_1 = \{ 1, 2 \}$, $A_2 = \{3, 4\}$
$B_1 = \{ 2, 5 \}$, $B_2 = \{3, 8\}$

$\cup_{i \in I}(A_i \setminus B_i) = \{1, 4 \}$
$\cup_{i \in I} A_i \setminus \cap_{i \in I} B_i = \{ 1, 2, 3, 4 \}$

Soln24

(a)

Suppose $x \in \cup_{i \in I}(A_i \cap B_i)$. Then there must exist atleast an $i = k$, such that $x \in A_k \land x \in B_k$. Since $x \in A_k$, it follows that $x \in \cup_{i \in I} A_i$. Similarly, since $x \in B_k$, it follows that $x \in \cup_{i \in I} B_i$. Thus we have $x \in \cup_{i \in I} A_i \land x \in \cup_{i \in I} B_i$, or $x \in (\cup_{i \in I} A_i) \cap (\cup_{i \in I} B_i)$. Since $x$ is arbitrary, it follows that $\cup_{i \in I}(A_i \cap B_i) \; \subseteq \; (\cup_{i \in I} A_i) \cap (\cup_{i \in I} B_i)$.

(b)

$A_1 = \{ 1, 2, 8 \}$, $A_2 = \{3, 4\}$
$B_1 = \{ 2, 5 \}$, $B_2 = \{3, 8\}$

$\cup_{i \in I}(A_i \cap B_i) = \{ 2, 3\}$
$(\cup_{i \in I} A_i) \cap (\cup_{i \in I} B_i) = \{ 2, 3, 8\}$

Soln25

Suppose $c = ab$, then both $a\,\vert\,c$ and $b\,\vert\,c$ are true.

Soln26

(a)

($\to$)Suppose $15\,\vert\,n$. Then $n = 15 \times k$ where $k \in \mathbb N$. It follows that $n = 3 \times 5k$, thus $3\,\vert\,n$. Similarly, $n = 5 \times 3k$, thus $5\,\vert\,n$.

($\leftarrow$)Suppose $3\,\vert\,n$ and $5\,\vert\,n$. Thus $n = 3k$ and $n = 5l$, where $k,l \in \mathbb N$. Thus $3k = 5l$. Since $k, l$ are natural numbers and $3$ and $5$ does not divide, it follows that $5\,\vert\,k$ and $3\,\vert\,l$. Or we can say that $5p = k$ and $3q = l$, where $p, q \in \mathbb N$. Thus we have $n = 3k = 15 p$ as well as $n = 5l = 15 q$. Thus $p = q$ and $n = 15p$. Thus we can conclude $15\,\vert\,n$.

(b)

Suppose $n = 90$, then $6\,\vert\,n$ as well as $10\,\vert\,n$ are true, but $60\,\vert\,n$ is not true.