Mathematics
Grade 10
15 min
Negations
Negations
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Introduction & Learning Objectives
Learning Objectives
Define a logical statement and its negation.
Determine the truth value of a simple statement and its negation.
Write the negation of a simple statement using appropriate language.
Write the negation of a quantified statement (e.g., 'all', 'some', 'none').
Apply the Law of Double Negation to simplify logical expressions.
Distinguish between a logical negation and a common opposite.
If your friend says, 'All dogs can fly,' what is the one specific thing you need to find to prove them wrong? 🐕🦺
This tutorial introduces the concept of negation, a fundamental building block of logic. You will learn how to correctly 'negate' or find the logical opposite of a statement, which is a critical skill for construct...
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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 triangle has three sides' is a true statement. 'What is your name?' is not a statement because it is a question.
Truth ValueThe classification of a statement as either true (T) or false (F).The truth value of the statement 'The Earth is flat' is false (F).
NegationThe logical opposite of a statement. If a statement is true, its negation is false. If a statement is false, its negation is true.The negation of 'The light is on' is 'The light is not on'.
Negation Symbol (~ or ¬)The symbol used in logic to represent negation. If 'p' is a statement, then '~p' (read as 'not p&#...
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Core Formulas
Rule of Negation
If the truth value of a statement `p` is T, then the truth value of `~p` is F. If the truth value of `p` is F, then the truth value of `~p` is T.
This is the fundamental rule defining the relationship between a statement and its negation. Their truth values are always opposite.
Law of Double Negation
~(~p) \equiv p
The negation of a negation is logically equivalent to the original statement. Saying 'It is not not raining' means the same thing as 'It is raining'.
Negating Universal Quantifiers
The negation of 'All A are B' is 'Some A are not B'.
To disprove a statement that claims something is true for ALL cases, you only need to find AT LEAST ONE case where it is not true.
Negating Existential Quantifiers...
5 more steps in this tutorial
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Challenging
Let p be the statement: 'For all regular polygons, the measure of an interior angle is less than 180 degrees.' This statement is true. Which of the following describes its negation, ~p?
A.~p is 'No regular polygons have an interior angle less than 180 degrees,' which is false.
B.~p is 'Some regular polygons do not have an interior angle less than 180 degrees,' which is false.
C.~p is 'Some regular polygons have an interior angle of exactly 180 degrees,' which is true.
D.~p is 'Some regular polygons do not have an interior angle less than 180 degrees,' which is true.
Challenging
The statement p is 'x² = 25'. If the negation, ~p, is true, which of the following is a possible value for x?
A.25
B.-5
C.5
D.Both B and C
Challenging
A student is asked to negate 'Some 3D figures have no vertices.' They write, 'Some 3D figures have vertices.' Why is this logically incorrect?
A.The student's answer is the opposite, not the negation.
B.The correct negation must start with 'All'.
C.The original statement is false.
D.The original statement and the student's answer can both be true simultaneously.
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