Truth values

Key Notes:

Definition of Truth Value:

A truth value tells whether a statement is true (T) or false (F).

Example:

  • Statement: “5 is greater than 2” → True
  • Statement: “The sky is green” → False

Types of Statements:

  • Declarative Statements: These are statements that can be either true or false.
  • Not Truth-Bearing Statements: Questions, commands, or opinions do not have a truth value.

Symbols Used:

  • T or 1 → for True
  • F or 0 → for False

Use in Logic:

Truth values are used in logic to analyze statements and arguments.

They help in evaluating logical connectives like:

  • AND (∧)
  • OR (∨)
  • NOT (¬)
  • IMPLIES (→)

Examples of Truth Values in Logical Statements:

  • “2 + 2 = 4” → True
  • “All birds can fly” → False (because some birds like ostriches cannot fly)

Negation and Truth Values:

The negation of a true statement is false, and the negation of a false statement is true.

Example:

  • Statement: “The earth is round” → True
  • Negation: “The earth is not round” → False

Truth Tables:

  • Truth tables are tools used to show all possible truth values of logical statements.
  • Useful for understanding compound statements and logical operators.

Learn with an example

✈️ If p is true, what is the truth value of ~p?

  • true
  • false
  • cannot be determined
  • Start by constructing a truth table that has only the row in which p is true.
p~p
T
  • Next, fill in the second column. Find the opposite of the value of p.
p~p
TF
  • Finally, look at the truth value in the ~p column. ~p is false.

✈️ If p is true and q is false, what is the truth value of q∨p?

  • true
  • false
  • cannot be determined
  • Start by constructing a truth table that has only the row in which p is true and q is false.
pqq∨p
TF
  • Next, fill in the third column. Determine if the value of either q or p is true.
pqq∨p
TFT
  • Finally, look at the truth value in the q∨p column. When p is true and q is false, q∨p is true.

✈️If p is true and q is true, what is the truth value of q∧p?

  • true
  • false
  • cannot be determined
  • Start by constructing a truth table that has only the row in which p is true and q is true.
pqq∧p
TT
  • Next, fill in the third column. Determine if the values of both q and p are true.
pqq∧p
TTT
  • Finally, look at the truth value in the q∧p column. When p is true and q is true, q∧p is true.

Let’s practice!🖊️