# Truth Tables

Truth table specifies the truth value of a compound proposition for all possible truth values of its constituent propositions.

## Negation(~)

If p is a statement variable, then negation of p,"not p", is denoted as "~p".

It has opposite truth value from p i.e., if p is true, then ~p is false; if p is false, then ~p is true.

Truth Table for ~p
p | ~p |

T | F |

F | T |

## Conjunction (∧)

Assume p and q are statements and the conjunction of p and q is "p and q", denoted as "p ∧q".

Note the following regarding the conjunction

- p∧q is true when both p and q are true.
- p∧q is false when either p or q is false, or both are false.

The Truth Table for p∧q is illustrated below:

Truth Table for p∧q
p | q | p∧q |

T | T | T |

T | F | F |

F | T | F |

F | F | F |

## Disjunction (∨) or Inclusive OR

Assume p and q are statements and the disjunction of p and q is "p or q", denoted as "p ∨q".

Note the following regarding the disjunction

- p∨q is true when either p or q is true, or both are true.
- p∨q is false when both p and q are false.

The Truth Table for p∨q is illustrated below:

Truth Table for p∨q
p | q | p∨q |

T | T | T |

T | F | T |

F | T | T |

F | F | F |

## Exclusive OR

When OR is used in its exclusive sense, the statement "p or q" mean "p or q but not both" or "p or q but not p and q"
which translates into symbols as (p∨q) ∧ ~(p∧q). It is abbreviated as p⊕q or p XOR q.
The Truth Table for p XOR q is illustrated below:

Truth Table for p⊕q
p | q | p⊕q |

T | T | F |

T | F | T |

F | T | T |

F | F | F |

## Conditional Statements

Consider the following statement: "If you get an A in Computer Science, then I will buy you your favorite phone."
This statement is composed of two simpler statements:

p:"You will get an A in Computer Science"

q:"I will buy you your favorite phone."

The original statement is then saying:

*if p is true, then q is true, or simply, if p, then q.*

This can also be expressed as: *p* **implies** *q*. It is denoted by **p → q.**

**p → q.** is false when p is true and q is false; otherwise it is true.

In **p → q.**, the statement **p** is called the **hypothesis (or antecedent)** and q is called the **conlcusion (or consequent)**

The Truth Table for p → q is illustrated below:

Truth Table for p → q
p | q | p→q |

T | T | T |

T | F | F |

F | T | T |

F | F | T |

## Biconditional

Assume p and q are statement variables, the biconditional of p and q is "p if and only if q".

it is denoted **p ↔ q.** *"if and only if"* is abbreviated as **iff.**

The symbol (double headed arrwo) "↔" is the **biconditional operator.**

Note the following regarding the biconditional

- p↔q is true when both p and q are true.
- p↔q is false when either p or q is false.

The Truth table for p ↔ q is illustrated below.

Truth Table for p↔q
p | q | p↔q |

T | T | T |

T | F | F |

F | T | F |

F | F | T |

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Date of last modification: February 27, 2019