Mathematics
Grade 10
15 min
Truth values
Truth values
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1
Introduction & Learning Objectives
Learning Objectives
Define a statement and its truth value.
Distinguish between statements, open sentences, and non-statements (like questions or commands).
Determine the truth value (True or False) of a simple statement.
Define and correctly apply the negation operator (~).
Determine the truth value of a negated statement.
Understand that a statement must be either true or false, but not both (Principle of Bivalence).
Is the statement 'Smartphones are smarter than humans' true or false? 🤔 Logic gives us the tools to analyze statements like this with precision!
In this tutorial, we will explore the most fundamental concept in logic: truth values. You will learn what makes a sentence a logical statement and how to assign it a value of 'True' or 'Fal...
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Key Concepts & Vocabulary
TermDefinitionExample
Statement (or Proposition)A declarative sentence that can be objectively classified as either true or false, but not both."A square has four equal sides." This is a statement because it is a declarative sentence and it is definitively true.
Truth ValueThe attribute of a statement that determines if it is true (T) or false (F).The truth value of the statement "5 is a prime number" is True (T).
Simple StatementA statement that expresses a single idea and does not contain any logical connectors."The Earth revolves around the Sun."
Open SentenceA sentence containing one or more variables, whose truth value depends on the value(s) assigned to the variables."x + 5 = 12". This is not a statement until a value is given for x. It's...
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Core Formulas
Truth Value Assignment
For any statement p, its truth value, V(p), must be either True or False. V(p) \in \{T, F\}
This is the fundamental rule that every statement you can evaluate has exactly one of two possible truth values. There is no 'in-between' or 'unknown' in classical logic.
The Rule of Negation
The truth value of the negation of p (denoted as ~p) is the opposite of the truth value of p. If V(p) = T, then V(~p) = F. If V(p) = F, then V(~p) = T.
Use this rule to find the truth value of a negated statement without re-evaluating the entire sentence. Simply flip the truth value of the original statement.
4 more steps in this tutorial
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Challenging
Let p be the statement: 'For any integer n, the number 2n+1 is odd.' What is the truth value of p?
A.False, because if n=0.5, 2n+1=2.
B.True, because multiplying any integer by 2 results in an even number, and adding 1 makes it odd.
C.False, because if n=0, 2n+1=1, which is not odd.
D.It is an open sentence, so it has no truth value.
Challenging
Consider the open sentence S: 'x² ≥ 25'. For which of these values of x is the NEGATION of S, ~S, a true statement?
A.x = 5
B.x = -5
C.x = 6
D.x = 4
Challenging
Which of the following is a statement in logic, even if its truth value is not immediately obvious or easy to find?
A.The 500th digit after the decimal point in the value of π is 7.
B.Mathematics is the most important school subject.
C.Find the value of π.
D.Let x be the 500th digit of π.
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