$$1. \sim A \vdash A \supset B \hspace{5em} (PMI)$$

$$2.\ \ \ A \vdash B \supset A \hspace{5em} (PMI)$$

Intuitively, by reference to the truth table, you can understand that they are true. For the first one, when $\sim A$ is true, $A$ is false and $A \supset B$ is true. For the second, when $A$ is true, $B \supset A$ is true. But this is semantic entailment, i.e there is no interpretation where the premises are true and the conclusion false. In SD, we need to prove these using inference rules.

We can prove these as follows.

$1. \sim A \vdash A \supset B$

$1. \sim A \hspace{10em} Assumption$

$ \ \ \ 2. \ A \hspace{10em} Assumption$

$ \ \ \ \ \ \ 3. \sim B \hspace{8em} Assumption$

$ \ \ \ \ \ \ 4.\ A \hspace{9em} 1\ Reiteration$

$ \ \ \ \ \ \ 5.\sim A \hspace{8em} 2 \ Reiteration$

$ \ \ \ 6.\ B \hspace{10em} 3-5 \sim E$

$7. \ A \supset B \hspace{10em} 2,6 \supset I$

Compare with (the content is the same as above but different display):

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```
1. ~A Assunmption
2.A Assumption
3.~B Assumption
4.A 1 Reiteration
5.~A 2 Reiteration
6.B 3-5 ~E
7. A ⊃ B 2,6 ⊃I
```

$2.\ A \vdash B \supset A$

$1.\ A \hspace{10em} Assumption$

$\ \ \ 2.\ B \hspace{9em} Assumption$

$\ \ \ 3.\ A \hspace{9em} 1,\ Reiteration$

$3.\ B \supset A \hspace{10em} 1,2 ⊃I$

Compare with (the content is the same thing as above):

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```
1. A Assumption
2. B Assumption
3. A 1 Reiteration
3.B ⊃ A 1,2 ⊃I
```

It is called LaTex notation. For example, the first deduction above looks like this in the post editor:

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```
$1. \sim A \hspace{10em} Assumption$
$ \ \ \ 2. \ A \hspace{10em} Assumption$
$ \ \ \ \ \ \ 3. \sim B \hspace{8em} Assumption$
$ \ \ \ \ \ \ 4.\ A \hspace{9em} 1\ Reiteration$
$ \ \ \ \ \ \ 5.\sim A \hspace{8em} 2 \ Reiteration$
$ \ \ \ 6.\ B \hspace{10em} 3-5 \sim E$
$7. \ A \supset B \hspace{10em} 2,6 \supset I$
```

$$\begin{vmatrix} 12 & 1 & 14 & 7 \\ 13 & 8 & 11 & 2 \\ 3 & 10 & 5 & 16 \\ 6 &15 & 4 & 9 \end{vmatrix}$$

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`$$\begin{vmatrix} 12 & 1 & 14 & 7 \\ 13 & 8 & 11 & 2 \\ 3 & 10 & 5 & 16 \\ 6 &15 & 4 & 9 \end{vmatrix}$$`

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`$\forall x \exists y (Lxy)$`

The syntax of the language is shown here: https://www.abisource.com/~msevior/math-reference.html

One just has to put this syntax between a single pair of dollar signs or a double pair of dollar signs. Integrals, summations, greek alphabets, and all mathematical symbols can be displayed nicely with this language and functionality. What do you think, isn't this cool?