# 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.

 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:

 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:

 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:

 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"

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:

 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.

 p q p↔q T T T T F F F T F F F T