Mathematics
Grade 10
15 min
Conditionals
Conditionals
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1
Introduction & Learning Objectives
Learning Objectives
Identify the hypothesis and conclusion of a conditional statement.
Write a conditional statement in 'if-then' form.
Determine the truth value of a conditional statement.
Write the converse, inverse, and contrapositive of a conditional statement.
Identify logically equivalent statements, specifically the conditional and its contrapositive.
Define and write a biconditional statement using 'if and only if'.
If you study for the test, then you will get a good grade. 🤔 What makes this statement a powerful tool in mathematics?
This lesson introduces conditional statements, the 'if-then' sentences that form the backbone of logical reasoning. We will explore how to break them down, rewrite them in different forms, and use them to b...
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Key Concepts & Vocabulary
TermDefinitionExample
Conditional StatementA logical statement with two parts, a hypothesis and a conclusion, typically written in 'if-then' form.If a figure is a square, then it is a rectangle.
HypothesisThe 'if' part of a conditional statement. It is the condition that is assumed to be true.In the statement 'If it is raining, then the ground is wet,' the hypothesis is 'it is raining'.
ConclusionThe 'then' part of a conditional statement. It is the result or outcome based on the hypothesis.In the statement 'If it is raining, then the ground is wet,' the conclusion is 'the ground is wet'.
ConverseA statement formed by switching the hypothesis and the conclusion of a conditional statement.The converse of 'If a figur...
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Core Formulas
Symbolic Notation for Conditionals
Let p be the hypothesis and q be the conclusion.
Conditional: p → q ('if p, then q')
Converse: q → p ('if q, then p')
Inverse: ~p → ~q ('if not p, then not q')
Contrapositive: ~q → ~p ('if not q, then not p')
This is the standard symbolic representation used in logic to analyze statements. The symbol '~' means 'not' (negation).
Rule of Logical Equivalence
A conditional statement and its contrapositive are logically equivalent. (p → q ↔ ~q → ~p)
The converse and the inverse are also logically equivalent. (q → p ↔ ~p → ~q)
Logically equivalent statements always have the same truth value. If the original conditional is true, its contrapositive must also be true. This is a powerful to...
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Challenging
If the conditional statement 'p → q' is known to be FALSE, what must be the truth values of p and q?
A.p is false and q is true.
B.p is true and q is false.
C.p is false and q is false.
D.p is true and q is true.
Challenging
The Isosceles Triangle Theorem states: 'If a triangle has two congruent sides, then it has two congruent angles.' The converse is also true. Which statement correctly expresses this as a true biconditional?
A.If a triangle has two congruent angles, then it is isosceles.
B.triangle is isosceles, and it has two congruent angles.
C.If a triangle does not have two congruent sides, then it does not have two congruent angles.
D.triangle has two congruent sides if and only if it has two congruent angles.
Challenging
If you are given that the contrapositive (~q → ~p) of a statement is FALSE, what can you conclude about the truth value of the inverse (~p → ~q)?
A.The inverse is true.
B.The inverse is false.
C.The truth value of the inverse cannot be determined.
D.The inverse is also false.
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