# How to Prove It - Solutions

## Chapter - 4, Relations

### Summary

• In last section, relations were described as subsets of cartesian products. Another alternative notation for relations can be:
• A point $(a,b)$ in a relation R, or $(a,b) \in R$, can be also be described by notation: $a R b$.
• This notation is similar to the way in mathematics where relationships between two objects are expressed by placing a symbol between these two objects. Eg: $x > y$, or $A \subseteq B$.
• Another way to see relations, is drawing their pictures. To describe a relation $R$ from $A$ to $B$. Two closed curves can be drawn adjacent to each other without touching. Draw points/vertices in one curve containing all the points from $A$ and similarly in other curve containing all points of $B$. Now if $x \in A$ and $y \in B$ and $(x,y) \in R$, then draw an edge pointing from $A$ to $B$.
• For relations like $R \subseteq A \times A$, directed graphs can be used to described the relation. Such relations may also be called as: $R$ is a relation on set $A$.
• Suppose $R$ is a relation on $A$, then:
• $R$ is reflexive on $A$ (or just reflexive, if $A$ is clear from context) if $\forall x \in A(x R x),$ or in other words $\forall x \in A((x,x) \in R)$.
• $R$ is symmetric if $\forall x \in A \forall y \in A(x R y \to y R x)$.
• $R$ is transitive if $\forall x \in A \forall y \in A \forall z \in A((x R y ∧ y R z) \to x R z)$.
• Suppose $A$ is a set. Let $i_A = \{(x, y) \in A \times A \, \vert \, x = y \}$. Then $i_A$ is called identity relation on $A$.
• Suppose $R$ is a relation on a set $A$, then:
• $R$ is reflexive iff $i_A \subseteq R$, where $i_A$ is the identity relation on A.
• $R$ is symmetric iff $R = R^{−1}$.
• $R$ is transitive iff $R \circ R \subseteq R$.

Skipping problems 1 to 3 as they need diagrams.

Soln4

(a) $\{ (a,c), (c,c), (d,a), (d,b), (b,d) \}$.

(b) $\{ (a,b), (b,a), (a,d), (b,d) \}$.

(c) $\{ (a,a), (b,b), (c,c), (d,d), (b,d), (d,b) \}$. Set is reflexive, symmetric and transitive.

(d) $\{ (a,b), (a,c), (a,d), (c,d), (b,d) \}$. Set is transitive.

Soln5 $\{ (a,y), (a,z), (b,x), (c,y), (c,z) \}$.

Soln6

$D_r \circ D_s = \{ (a,c) \, \vert \, \exists b((a,b) \in D_s \land (b,c) \in D_r) \}$.
$D_r \circ D_s = \{ (a,c) \, \vert \, \exists b( \vert a - b \vert < s \land \vert b - c \vert < r ) \}$.

Since $\vert a - c \vert = \vert (a-b) + (b-c) \vert$. Thus by triangular inequality, we have:

$\vert a - c \vert < \vert a-b \vert + \vert b-c \vert$.
Since $\vert a - b \vert < s$ and $\vert b - c \vert < r$, we have:
$\vert a - c \vert < r + s$.

Thus we have,
$D_r \circ D_s = \{ (a,c) \, \vert \, \vert a - c \vert < r + s \}$.

Soln7

($\to$)Suppose $R$ is reflexive. Suppose $(x,x) \in i_A$. Since $i_A$ is a relation on $A$, it follows that $x \in A$. Since $R$ is reflexive and $R$ is defined on $A$, it follows that $(x,x) \in R$. Thus if $(x,x) \in i_A$, then $(x,x) \in R$. Since $(x,x)$ is arbitrary, we can conclude that $i_A \subseteq R$.

($\leftarrow$)Suppose $i_A \subseteq R$. Suppose $x \in A$. Since $x \in A$, $(x,x) \in i_A$. Since $i_A \subseteq R$, it follows that $(x,x) \in R$. Thus if $x \in A$, then $(x,x) \in R$. Since $x$ is arbitrary, we can conclude that $\forall x \in A((x,x) \in R)$, which is equivalent to saying that $R$ is reflexive.

Thus from both directions we can conclude that: $R$ is reflexive, iff, $i_A \subseteq R$.

Soln8

($\to$)Suppose $R$ is transitive. Suppose $(a,c) \in R \circ R$. Thus there exist an element $b \in A$ such that $(a,b) \in R$ and $(b,c) \in R$. Now since $R$ is transitive and $(a,b) \in R \land (b,c) \in R$, it follows that $(a,c) \in R$. Thus if $(a,c) \in R \circ R$, then $(a,c) \in R$. Since $(a,c)$ is arbitrary, we can conclude that $R \circ R \subseteq R$.

($\leftarrow$)Suppose $R \circ R \subseteq R$. Suppose $a,b,c$ are three elements in $A$ such that $(a,b) \in R \land (b,c) \in R$. Now since $(a,b) \in R$ and $(b,c) \in R$, it follows $(a,c) \in R \circ R$. Since $R \circ R \subseteq R$, it follows that $(a,c) \in R$. Since $a,b,c$ are arbitrary, we can conclude that $\forall a \in A \forall b \in A \forall c \in A((a,b) \in R \land (b,c) \in R \to (a,c) \in R)$, which is equivalent to saying that $R$ is transitive.

Soln9

(a)

($\to$)Suppose $(a,c) \in R \circ i_A$. Thus there exist an element $b \in A$ such that $(a,b) \in i_A \land (b,c) \in R$. Since $(a,b) \in i_A$, it follows $a = b$. Thus $(b,c) \in R$ is equivalent to $(a,c) \in R$. It follows that if $(a,c) \in R \circ i_A$, then $(a,c) \in R$. Since $(a,c)$ is arbitrary, we can conclude that $R \circ i_A \subseteq R$.

($\leftarrow$)Suppose $(a,c) \in R$. Thus $a \in A \land c \in B$. Since $a \in A$, it follows that $(a,a) \in i_A$. Now since $(a,a) \in i_A$ and $(a,c) \in R$, it follows that $(a,c) \in R \circ i_A$. Thus if $(a,c) \in R$, then $(a,c) \in R \circ i_A$. Since $(a,c)$ is arbitrary, we can conclude that $R \subseteq R \circ i_A$.

Thus we have $R \circ i_A \subseteq R$ as well as $R \subseteq R \circ i_A$. It follows that $R = R \circ i_A$.

(b)

($\to$)Suppose $(a,c) \in i_B \circ R$. Thus there exist an element $b \in B$ such that $(a,b) \in R \land (b,c) \in i_B$. Since $(b,c) \in i_B$, it follows that $b = c$. Thus $(a,b) \in R$ is equivalent to saying that $(a,c) \in R$. Thus if $(a,c) \in i_B \circ R$, then $(a,c) \in R$. Since $(a,c)$ is arbitrary, we can conclude that $i_B \circ R \subseteq R$.

($\leftarrow$)Suppose $(a,b) \in R$. Thus $a \in A \land b \in B$. Since $b \in B$, it follows that $(b,b) \in i_B$. Now since $(a,b) \in R$ and $(b,b) \in i_B$, it follows that $(a,b) \in i_B \circ R$. Thus if $(a,b) \in R$, then $(a,b) \in i_B \circ R$. Since $(a,b)$ is arbitrary, we can conclude that $R \subseteq i_B \circ R$.

Thus we have $i_B \circ R \subseteq R$ as well as $R \subseteq i_B \circ R$. It follows that $R = i_B \circ R$.

Soln10

• $i_D \subseteq S^{-1} \circ S$
Suppose $(x,x) \in i_D$. Thus $x \in D$, or $Dom(S)$. Since $x \in Dom(S)$, there must exist an element $y$ such that $(x,y) \in S$. It follows that $(y,x) \in S^{-1}$. Since $(x,y) \in S$ and $(y,x) \in S^{-1}$, it follows that $(x,x) \in S^{-1} \circ S$. Thus if $(x,x) \in i_D$, then $(x,x) \in S^{-1} \circ S$. Since $(x,x)$ is arbitrary, we can conclude that $i_D \subseteq S^{-1} \circ S$.

• $i_R \subseteq S \circ S^{−1}$
Suppose $(b,b) \in i_R$. Thus $b \in R$, or $Ran(S)$. Since $b \in Ran(S)$, there must exist an element $a$ such that $(a,b) \in S$. It follows that $(b,a) \in S^{-1}$. Since $(a,b) \in S^{-1}$ and $(a,b) \in S$, it follows that $(b,b) \in S \circ S^{-1}$. Thus if $(b,b) \in i_D$, then $(b,b) \in S \circ S^{-1}$. Since $(b,b)$ is arbitrary, we can conclude that $i_R \subseteq S \circ S^{-1}$.

Soln11

Suppose $(x,y) \in R$. Thus $x \in A \land y \in A$. Since $R$ is reflexive and $y \in A$, it follows $(y,y) \in R$. Now since $(x.y) \in R$ and $(y,y) \in R$, it follows $(x,y) \in R \circ R$. Thus if $(x,y) \in R$, then $(x,y) \in R \circ R$. Since $(x,y)$ is arbitrary, it follows that $R \subseteq R \circ R$.

Soln12

(a) Suppose $R$ is reflexive. Suppose $(x,y) \in i_A$. Thus $x = y$, and we can also say that $(y,x) \in i_A$. Since $R$ is reflexive, $i_A \subseteq R$. Thus $(x,y) \in R$. Since $(x,y) \in R$, it follows $(y,x) \in R^{-1}$. Thus we have: if $(y,x) \in i_A$, then $(y,x) \in R^{-1}$. Or we can conclude that $i_A \subseteq R^{-1}$, which is equivalent to saying that $R^{-1}$ is reflexive.

(b) Suppose $R$ is symmetric. Suppose $(x,y) \in R^{-1}$. It follows that $(y,x) \in R$. Since $R$ is symmetric, it follows that $(x,y) \in R$. Now since $(x,y) \in R$, it follows $(y,x) \in R^{-1}$. Thus if $(x,y) \in R^{-1}$, then $(y,x) \in R^{-1}$. Since $(x,y)$ is arbitrary, we can conclude that $R^{-1}$ is symmetric.

(c) Suppose $R$ is transitive. Suppose $(a,c) \in R^{-1}$. It follows that $(c,a) \in R$. Since $R$ is transitive, it follows that $\exists b \in A ((c,b) \in R \land (b,a) \in R)$. Thus we have $(a,b) \in R^{-1}$ and $(b,c) \in R^{-1}$. Now if $(a,c) \in R^{-1}$, then $\exists b \in A((a,b) \in R^{-1} \land (b,c) \in R^{-1}$. Since $(a,c)$ is arbitrary, we can conclude that $R$ is transitive.

Soln13

(a) True. Suppose $R_1$ and $R_2$ are reflexive. Suppose $(x,y) \in i_A$. Since $R_1$ is reflexive, it follows $(x,y) \in R_1$. Thus we can also say $(x,y) \in R_1 \cup R_2$. It follows that if $(x,y) \in i_A$, then $(x,y) \in R_1 \cup R_2$. Since $(x,y)$ is arbitrary, we can conclude that $i_A \subseteq R_1 \cup R_2$. Thus $R_1 \cup R_2$ is reflexive.

(b) True. Suppose $R_1$ and $R_2$ are symmetric. Suppose $(x,y) \in R_1 \cup R_2$. Thus we have two cases:

• Case 1: $(x,y) \in R_1$
Since $R_1$ is symmetric, it follows $(y,x) \in R_1$. Thus we can also say $(y,x) \in R_1 \cup R_2$.

• Case 2: $(x,y) \in R_2$
Since $R_2$ is symmetric, it follows $(y,x) \in R_2$. Thus we can also say $(y,x) \in R_1 \cup R_2$.

Thus from both cases $(y,x) \in R_1 \cup R_2$. Since $(x,y)$ is arbitrary, we can conclude that $R_1 \cup R_2$ is symmetric.

(c) False. Counter Example: $A = \{1, 2, 3 \}, R_1 = \{ \{1,2\}, \{2, 2\} \}, R_2 = \{ \{2,3\} \}$. Clearly $R_1$ and $R_2$ are transitive but $R_1 \cup R_2$ is not transitive. Because $\{1,2\} \in R_1 \cup R_2$ and $\{2,3\} \in R_1 \cup R_2$ but $\{1,3\} \notin R_1 \cup R_2$.

Soln14

(a) True. Suppose $R_1$ and $R_2$ are reflexive. Suppose $x \in A$. Since $R_1$ is reflexive, it follows that $(x,x) \in R_1$. Similarly, since $R_2$ is reflexive, it follows that $(x,x) \in R_2$. Thus $(x,x) \in R_1 \cap R_2$. Thus if $x \in A$ then $(x,x) \in R_1 \cap R_2$. Since $x$ is arbitrary, we can conclude that $R_1 \cap R_2$ is reflexive.

(b) True. Suppose $R_1$ and $R_2$ are symmetric. Suppose $(x,y) \in R_1 \cap R_2$. Thus $(x,y) \in R_1$. Since $R_1$ is symmetric, it follows that $(y,x) \in R_1$. Similarly, since $(x,y) \in R_2$ and $R_2$ is symmetric, it follows that $(y,x) \in R_2$. Thus we have $(y,x) \in R_1 \cap R_2$. Since $(x,y)$ is arbitrary, we can conclude that $R_1 \cap R_2$ is symmetric.

(c) True. Suppose $R_1$ and $R_2$ are transitive. Suppose $a,b,c$ are elements in $A$ such that $(a,b) \in R_1 \cap R_2$ and $(b,c) \in R_1 \cap R_2$. Thus $(a,b) \in R_1 \land (b,c) \in R_1$ and $(a,b) \in R_2 \land (b,c) \in R_2$. Since $R_1$ is transitive, it follows that $(a,c) \in R_1$. Similarly since $R_2$ is transitive, it follows that $(a,c) \in R_2$. Thus $(a,c) \in R_1 \cap R_2$. Since $a,b,c$ are arbitrary, we can conclude that $R_1 \cap R_2$ is transitive.

Soln15

(a) False. $A = \{1\}, R_1 = \{ \{1,1\} \}, R_2 = \{ \{1,1\} \}$. Thus $R_1 \setminus R_2 = \phi$. Clearly, R_2 is not reflexive, since $1 \in A$ but $(1,1) \notin R_1 \setminus R_2$.

(b) True. Suppose $R_1$ and $R_2$ are symmetric. Suppose $(x,y) \in R_1 \setminus R_2$. Thus $(x,y) \in R_1$. Since $R_1$ is symmetric, it follows that $(y,x) \in R_1$. Similarly, since $(x,y) \notin R_2$ and $R_2$ is symmetric, it follows that $(y,x) \notin R_2$. Thus we have $(y,x) \in R_1 \setminus R_2$. Since $(x,y)$ is arbitrary, we can conclude that $R_1 \cap R_2$ is symmetric.

(c) False. Counter Example: $A = \{1, 2, 3 \}, R_1 = \{ \{1,2\}, \{2, 3\}, \{1, 3\} \}, R_2 = \{ \{1,3\} \}$. Thus $R_1 \setminus R_2 = \{ \{1,2\}, \{2, 3\} \}$.
Clearly $R_1$ and $R_2$ are transitive but $R_1 \setminus R_2$ is not transitive. Because $\{1,2\} \in R_1 \setminus R_2$ and $\{2,3\} \in R_1 \setminus R_2$ but $\{1,3\} \notin R_1 \setminus R_2$.

Soln16

Suppose $R$ and $S$ are reflexive. Suppose $x \in A$. Since $R$ is reflexive, $(x,x) \in R$. Similarly, since $S$ is reflexive, $(x,x) \in S$. Since $(x,x) \in S$ and $(x,x) \in R$, it follows that $(x,x) \in R \circ S$. Thus if $x \in A$, then $(x,x) \in R \circ S$. Since $x$ is arbitrary, we can conclude that $R \circ S$ is reflexive.

Soln17

Suppose $R$ and $S$ are symmetric.

($\to$)Suppose $R \circ S$ is symmetric. Suppose $(x, z) \in R \circ S$.

Since $R \circ S$ is symmetric, $(x, z) \in R \circ S$ iff
$(z, x) \in R \circ S$ iff
$\exists y \in A((z,y) \in S \land (y,x) \in R$. iff
Since $S$ and $R$ are symmetric,
$\exists y \in A((y,z) \in S \land (x,y) \in R$. iff
$\exists y \in A((x,y) \in R \land (y,z) \in S$. iff
$(x,z) \in S \circ R$

Thus if $(x,z) \in R \circ S$, iff $(x,z) \in S \circ R$. Since $(x,z)$ is arbitrary, we can conclude that $R \circ S = S \circ R$.

($\leftarrow$)Suppose $R \circ S = S \circ R$. Suppose $(x,z) \in S \circ R$. Since $(x,z) \in S \circ R$, there must exist an element $y \in A$ such that $(x,y) \in R \land (y,z) \in S$. Since $R$ and $S$ are symmetric, it follows that $(z,y) \in S$ and $(y,x) \in R$. It follows that $(z,x) \in R \circ S$. Since $R \circ S = S \circ R$, it follows $(z,x) \in S \circ R$. Thus if $(x,z) \in S \circ R$, then $(z,x) \in S \circ R$. Since $(x,z)$ is arbitrary, we can conclude that $S \circ R$ is symmetric.

Soln18

Suppose $R$ and $S$ are transitive. Suppose $S \circ R \subseteq R \circ S$. Suppose $x,y,z$ are elements in $A$ such that $(x,y) \in R \circ S$ and $(y,z) \in R \circ S$. Since $(x,y) \in R \circ S$, it follows $\exists p \in A((x,p) \in S \land (p,y) \in R$.
Similarly since $(y,z) \in R \circ S$, it follows $\exists q \in A((y,q) \in S \land (q,z) \in R$. Since $(p,y) \in R$ and $(y,q) \in S$, it follows that $(p,q) \in S \circ R$. Since $S \circ R \subseteq R \circ S$, it follows $(p,q) \in R \circ S$.
Since $(p,q) \in R \circ S$, it follows that $\exists m \in A((p,m) \in S \land (m,q) \in R)$.

Since $(x,p) \in S$ and $(p,m) \in S$ and since $S$ is transitive, it follows $(x,m) \in S$.
Similarly, since $R$ is transitive, and $(m,q) \in R \land (q,z) \in R$, it follows $(m,z) \in R$.

Thus since $(x,m) \in S \land (m,z) \in R$, it follows that $(x,z) \in R \circ S$. Thus if $(x,y) \in R \circ S \land (y,z) \in R \circ S$, then $(x,z) \in R \circ S$. Since $x,y,z$ are arbitrary, it follows that $R \circ S$ is transitive.

Soln19

(a)

Proof and Theorem both are not correct. Problem with proof is, it is not using the existential instantiation properly. In the proof, if $(X,Y) \in S$, then there exists $x$ and $y$ such that $x \in X$ and $y \in Y$. Also if $(Y,Z) \in S$, then there exists $y$ and $z$ such that $y \in Y$ and $z \in Z$. Now here former $y$ may not be equal to latter $y$. But in the proof, it was wrongly assumed.

(b)

(Update: 3rd June ‘18, Earlier solution contained mistakes. Thanks William for pointing out.)

Counter example:

$A = \{ 1, 2, 3 \}$.
$R = \{ \{1, 2\}, \{2, 2\}, \{3, 3\} \}$.

$\{ \{1,2\}, \{2,3\} \} \in S$, since $1 \in \{1,2\} \land 2 \in \{2,3\}$ and $1 R 2$.
$\{ \{2,3\}, \{1,3\} \} \in S$, since $3 \in \{2,3\} \land 3 \in \{1,3\}$ and $3 R 3$.

But $\{ \{1,2\}, \{1,3\} \} \notin S$.

Earlier incorrect solution

Counter example:

$A = \{ 1, 2, 3 \}$.
$R = \{ \{1, 2\}, \{2, 2\}, \{3, 3\} \}$.

$\{ \{1,2\}, \{2,3\} \} \in S$, since $1 \in \{1,2\} \land 2 \in \{2,3\}$ and $1 R 2$.
$\{ \{2,3\}, \{1,3\} \} \in S$, since $2 \in \{2,3\} \land 1 \in \{1,3\}$ and $1 R 2$.

But $\{ \{1,2\}, \{1,3\} \} \notin S$.

Soln20

Suppose $R$ is transitive. Suppose $(X,Y) \in S$ and $(Y, Z) \in S$. Since $X,Y, Z$ are not empty, suppose $x \in X$, $y \in Y$, $z \in Z$ are arbitrary elements. Then by the definition of $S$, $xRy$ and $yRz$. Since $xRy$, $yRz$, and $R$ is transitive, $xRz$. But then since $x \in X$ and $z \in Z$, it follows from the definition of $S$ that $(X, Z) \in S$. Thus, $S$ is transitive.

Consider the case that $B$ may contain empty sets. Suppose $(X,Y) \in S$, then $\forall x \in X \forall y \in Y(xRy)$. It follows $\forall x( x \in X \to \forall y(y \in Y \to xRy))$. This can be true if $X = \phi \lor Y = \phi$. Thus we can have the following counter example:

Suppose $X \ne \phi$ and $Y = \phi$ and $Z \ne \phi$, also suppose there is an element $x \in X$ and an element $z \in Z$ such that $(x,z) \notin R$.

Since $Y = \phi$, $(X,Y) \in S$. Similarly $(Y,Z) \in S$. But since $(x,z) \notin R$, it follows $(X,Z) \notin S$. Thus $S$ is not transitive if $B$ contains empty sets.

Soln21

(a) True. Suppose $R$ is reflexive. Suppose $X \in P(A)$. Suppose $x \in X$. Thus there exist an element $y \in X$ such that $y = x$. Thus $x = y$ and since $R$ is reflexive, $xRy$. Since $x$ is arbitrary, we can say that $\forall x \in X \exists y \in X(xRy)$. It follows that $(X,X) \in S$. Thus if $X \in P(A)$, then $(X,X) \in S$. Thus $S$ is reflexive.

(b) False. Counter example:

$A = \{ 1, 2, 3 \}, R = \{ \{1,2\}, \{2,1\}, \{2,2\} \}$.

Now we can easily see that $( \{1,2\}, \{2,3\} ) \in S$ but $( \{2,3\}, \{1,2\} ) \notin S$.

(c) True. Suppose $R$ is transitive. Suppose $(X,Y) \in S \land (Y,Z) \in S$. Since $(X,Y) \in S$, there exist an element $y_0 \in Y$ such that $\forall x \in X(xRy_0)$. Also since $(Y,Z) \in S$, there exist an element $z_0 \in Z$ such that $\forall y \in Y(yRz_0)$. Thus $y_0Rz_0$. Since $R$ is transitive, it follows that $\forall x \in X(xRz_0)$. Thus $(X,Z) \in S$. Since $X,Y,Z$ are arbitrary, it follows that $S$ is transitive.

Soln22 Proof and theorem both are false. The flaw in the proof is there may not exist any element $y$ such that $(x,y) \in R$. Counter example:

$A = \{1,2,3\}, R = \{ \{1,1\}, \{1,2\}, \{2,1\}, \{2,2\} \}$. Thus $R$ is symmetric and transitive. But since $\{3,3\} \notin R$, it is not reflexive.

Soln23

Suppose $aRb$ and $bRc$. Suppose $X \subseteq A \setminus \{a,c\}$. To prove $R$ is transitive,we need to prove $aRc$. To prove $aRc$, we have to prove that if $X \cup \{a\} \in \mathcal F$, then $X \cup \{c\} \in \mathcal F$. Suppose $X \cup \{a\} \in \mathcal F$. Now consider two exhaustive cases:

Case 1: $b \notin X$. Thus $X \subseteq A \setminus \{a,b,c \}$. Since $aRb$ and $X \cup \{a\} \in \mathcal F$, it follows that $X \cup \{b\} \in \mathcal F$. Also, since $bRc$, and $X \cup \{b\} \in \mathcal F$, it follows that $X \cup \{c\} \in \mathcal F$.

Case 2: $b \in X$. Suppose $X' = (X \cup \{a\}) \setminus \{b\}$. Now since $b \in X$ and $X' = (X \cup \{a\}) \setminus \{b\}$, it follows that $X' \cup b = X \cup \{a\}$. Since $X \cup \{a\} \in \mathcal F$, it follows $X' \cup b \in \mathcal F$. Since $bRc$ and $X' \cup b \in \mathcal F$, it follows that $X' \cup \{c\} \in \mathcal F$. Since $X' = (X \cup \{a\}) \setminus \{b\}$, it follows $((X \cup \{a\}) \setminus \{b\}) \cup \{c\} \in \mathcal F$, or $(X \cup \{a\} \cup \{c\}) \setminus \{b\} \in \mathcal F$.
Now consider $X'' = (X \cup \{c\})\setminus \{b\}$. Thus $X'' \cup \{a\} = ((X \cup \{c\})\setminus \{b\}) \cup \{a\} = (X \cup \{a\} \cup \{c\}) \setminus \{b\}$. But $(X \cup \{a\} \cup \{c\}) \setminus \{b\} \in \mathcal F$, it follows that $X'' \cup \{a\} \in \mathcal F$. Since $a \notin X$, it follows $a \notin X''$. Thus we have $a \notin X'' \land b \notin X''$ and $X'' \subseteq A$ and $X'' \cup \{a\} \in \mathcal F$, it follows that $X'' \cup \{b\} \in \mathcal F$. Since $X'' = (X \cup \{c\})\setminus \{b\}$ and $b \in X$, it follows that $X \cup \{c\} = X'' \cup \{b\}$. Since $X \cup \{c\} = X'' \cup \{b\}$ and $X'' \cup \{b\} \in \mathcal F$, it follows that $X \cup \{c\} \in \mathcal F$.

Thus from both cases $X \cup \{c\} \in \mathcal F$.