Practice Problems -   -  Use 6 Dot Entry   Switch to Nemeth Tutorial

# Lesson 8.2: Conjunction, Disjunction, Conditionals, and Biconditionals

## Symbols

$\wedge \phantom{\rule{.3em}{0ex}}\text{conjunction}$
⠈⠦

$\vee \phantom{\rule{.3em}{0ex}}\text{disjunction}$
⠈⠖

$\sim \phantom{\rule{.3em}{0ex}}\text{tilde}$
⠈⠔

$\to \phantom{\rule{.3em}{0ex}}\text{right pointing arrow}$
⠳⠕

$↔\phantom{\rule{.3em}{0ex}}\text{two-way arrow}$
⠳⠺⠗⠕

## Review

Arrow symbols consist of a shaft and a barbed tip. The barbed tip can point in any direction and be full, half, curved or directionless. The shaft can take many forms (solid, double, curved, dotted, etc.). An arrow symbol is formed with the opening arrow indicator and a closing symbol with dots arranged in a consistent pattern to indicate the direction of the arrow. Additional symbols are used to indicate the type of barbed tip and its direction when the arrow shape has unusual tips or shafts. Symbols for arrows with unusual shafts and a standard barbed tip or arrows with unusual tips can be found in Guidelines for Technical Material (2014) §13. A grade 1 indicator must be used with the opening arrow indicator since it also has a grade 2 meaning unless grade 1 mode has already been set.

## Explanation

Conjunction and disjunction are types of compound statements used in logic functions. They are created when two statements are joined using logical connectors or operators. A conjunction is a compound statement that is formed by joining two statements with the "and" logical operator ∧, (inverted v shape). A conjunction implies that both statements are true. The conjunction symbol is two cells, dot four in the first cell and dots two three six in the second cell.

A disjunction is a compound statement formed by joining two statements with the "or" logical operator ∨, (upright v shape). A disjunction implies that at least one statement is true. The symbol is two cells, dot four in the first cell and dots two three five in the second cell.

### Example 1

$x\wedge y$
⠭⠈⠦⠽

### Example 2

$x\vee y$
⠭⠈⠖⠽

### Example 3

$x\vee \left(z\wedge y\right)$
⠭⠈⠖⠐⠣⠵⠈⠦⠽⠐⠜

### Example 4

$\left(4<5\right)\wedge \left(5>4\right)$
⠐⠣⠼⠙⠀⠈⠣⠀⠼⠑⠐⠜⠈⠦⠐⠣⠼⠑⠀⠈⠜⠀⠼⠙⠐⠜

The tilde is used to represent negation of a statement. The tilde is two cells, dot four in the first cell and dots three five in the second.

### Example 5

$\sim x\wedge y$
⠈⠔⠭⠈⠦⠽

### Example 6

$r\wedge \sim s$
⠗⠈⠦⠈⠔⠎

### Example 7

$\sim \left(r\vee s\right)$
⠈⠔⠐⠣⠗⠈⠖⠎⠐⠜

A conditional statement is an if-then statement (if x then y) in which x is a hypothesis and y is a conclusion. In its converse, the hypothesis and conclusion exchange places (if y then x). A conditional statement is symbolized by use of a one-way pointing arrow →, dots one two five six, dots one three five. An arrow is considered a sign of comparison and is spaced as such. The arrow indicator requires grade 1 mode. In Examples 8 and 9, both x and y are standing alone and both require grade 1 mode. The grade 1 passage indicator is used to turn on grade 1 mode for the entire expression.

### Example 8

$x\to y$
⠰⠰⠰⠭⠀⠳⠕⠀⠽⠰⠄

### Example 9

$y\to x$
⠰⠰⠰⠽⠀⠳⠕⠀⠭⠰⠄

A biconditional statement is a combination of a conditional statement and its converse, written in the if and only if form (x if and only if y). The biconditional is symbolized by a simple two-way arrow ↔ which requires four cells: the arrow indicator, dots one two five six, dots two four five six and dots one two three five for the left and right pointing barbs, and the closing indicator, dots one three five. Grade 1 passage indicators are used with Examples 10 and 11.

### Example 10

$x↔y$
⠰⠰⠰⠭⠀⠳⠺⠗⠕⠀⠽⠰⠄

### Example 11

$x↔y=\left(x\to y\right)\wedge \left(y\to x\right)$
⠰⠰⠰⠭⠀⠳⠺⠗⠕⠀⠽⠀⠐⠶⠀⠐⠣⠭⠀⠳⠕⠀⠽⠐⠜⠈⠦⠐⠣⠽⠀⠳⠕⠀⠭⠐⠜⠰⠄

### Example 12

$x\to \sim y$
⠰⠰⠰⠭⠀⠳⠕⠀⠈⠔⠽⠰⠄