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# Lesson 10.2: Conjuction, Disjunction, Conditionals, and Biconditionals

## Logical Operators - Compound Statements

There are two types of compound statements used in logic functions. These are called conjunction and disjunction. A conjunction implies that both statements are true, while disjunction implies that at least one statement is true. With a conjunction, statements are connected by the word "and" while with disjunction statements are connected by the word "or."

The symbol for conjunction is ∧. This is a two-cell symbol of dot four followed by dots one four six.
The symbol for disjunction is ∨. This is also a two-cell symbol of dot four followed by dots three four six.

## Rules for these symbols are similar to other signs of comparison.

If a tilde appears in front of a statement, this indicates to find the complement of the statement.

The conjunction and disjunction symbols are considered operations. Thus, there is no space before or after the symbol. It is spaced similarly to a plus or minus sign.

If the conjunction or disjunction signs have other symbols such as bars or dots superscribed or subscribed, the symbol is then considered a sign of comparison.

## Biconditionals (signs of shape: arrows)

Arrows are used for a variety of purposes in mathematics. Examples of these include use as signs of comparison, it follows that, approaches, or yields, as super-positioned symbols to indicate rays or vectors, or as interior symbols within other shapes. When constructing arrows, the shape indicator is brailled first. The arrows are then transcribed in the order in which their symbols appear, that is, the vertical direction if appropriate, the shaft, or the arrowhead. A left-pointing arrow would have the arrowhead brailled after the shape indicator, followed by the shaft. A right-pointing arrow would have the shaft brailled first. Arrows pointing in directions other than left or right require the appropriate arrow direction indicator following the shape indicator, as listed below.

1. directly-over indicator, dots one two six, for up-pointing arrows
2. directly-under indicator, dots one four six, for down-pointing arrows
3. superscript indicator for arrows pointing up to the left or up to the right
4. subscript indicator for arrows pointing down to the left or down to the right.

The shaft of the arrow, when it is displayed, is composed of two cells of dots two five for a single-shaft arrow, and two cells of dots two three five six for an arrow with a double shaft. The right contracted arrow, the arrowhead with no shaft, is to be used when a right-pointing arrow appearing in regular type font, having a full barb with a single shaft of ordinary length, occurs by itself. If the right-pointing arrow appears in a different type font, for example, boldface or italic, or has a different type of barb, curved, blunted, or half-barbed, the contracted form is not to be used. The contracted form is also not used if the right-pointing arrow has a shaft of a different length than other arrows in the material, or has a double or triple shaft.

Listed below are some of the common arrows.

full-barbed arrows:

$\text{right-pointing, contracted form →}$
⠫⠕

$\text{right-pointing, uncontracted form, single shaft →}$
⠫⠒⠒⠕

$\text{left-pointing, single shaft ←}$
⠫⠪⠒⠒

$\text{double-barb at both ends ↔}$
⠫⠪⠒⠒⠕

$\text{right-pointing, double shaft ⇒}$
⠫⠶⠶⠕

$\text{left-pointing, double shaft ⇐}$
⠫⠪⠶⠶

$\text{down-pointing, single shaft ↓}$
⠫⠩⠒⠒⠕

$\text{up-pointing, single shaft ↑}$
⠫⠣⠒⠒⠕

Unlike the logical operators, arrows are considered signs of comparison and are spaced accordingly. This lesson will focus on the use of arrows primarily in conditional and biconditional statements. If you have two statements p and q, they can be combined to make a conditional statement such as If p, then q. The converse of this statement would then be If q, then p. If both conditional statements are true, they can be combined to create a biconditional that would be worded "p if and only if q". A tilde may be used in a conditional statement to show negation such as not p or not q.

### Example 1

$p\wedge q$
⠏⠈⠩⠟

### Example 2

$p\vee q$
⠏⠈⠬⠟

### Example 3

$p\vee \left(r\wedge q\right)$
⠏⠈⠬⠷⠗⠈⠩⠟⠾

### Example 4

$\left(5<6\right)\wedge \left(3>1\right)$
⠷⠢⠀⠐⠅⠀⠼⠖⠾⠈⠩⠷⠒⠀⠨⠂⠀⠼⠂⠾

### Example 5

$\sim p\wedge \sim q$
⠈⠱⠏⠈⠩⠈⠱⠟

### Example 6

$\sim \left(p\vee q\right)$
⠈⠱⠷⠏⠈⠬⠟⠾

### Example 7

$p\to q$
⠏⠀⠫⠒⠒⠕⠀⠟

### Example 8

$q\to p$
⠟⠀⠫⠒⠒⠕⠀⠏

### Example 9

$p↔q$
⠏⠀⠫⠪⠒⠒⠕⠀⠟

### Example 10

$p\to \sim q$
⠏⠀⠫⠒⠒⠕⠀⠈⠱⠟