Conditional Statements
Vocabulary
Conditional Statement
• A logical statement that has two parts, a
hypothesis and a conclusion
• EX: All cats are mammals
If-then Form
• A form of a conditional statement in which the
“if” part contains the hypothesis and the
“then” part contains the conclusion.
• EX: If an animal is a cat, then it is a mammal.
• Truth Value: true!
Hypothesis
• The “if” part of a conditional statement
• EX: If an animal is a cat, then it is a mammal.
Conclusion
• The “then” part of a conditional statement.
• EX: If an animal is a cat, then it is a mammal.
Converse
• Switch the hypothesis and conclusion.
• EX: If an animal is a mammal, then it is a cat.
• Truth Value: not true.
Inverse
• Negate the hypothesis and conclusion
• (keep them IN order, but add a “not”)
• EX: If an animal is NOT a cat, then it is NOT a
mammal.
• Truth value: not true (counterexample: dog)
Contrapositive
• Switch the order AND negate
• EX: If an animal is not a mammal then it is not
a cat.
• Truth value: true!
Biconditional Statement
• A statement that contains the phrase “if and
only if” in between the hypothesis and
conclusion.
• EX: An animal is a cat if and only if it is a
mammal.
• Truth value: NOT true, so we can’t write this
as a biconditional.
The car runs when there is gas in the tank.
• If-then form:
If the car runs, then there is gas in the tank (T)
• Converse: (switch order)
If there is gas in the tank, then the car runs (F)
• Inverse: (IN order, NOT)
If the car does NOT run, then there is NO gas (F)
• Contrapositive: (switch, NOT)
If there’s no gas, the car won’t run (T)
• Biconditional(?):
All squares are quadrilaterals
• If-then form:
• Converse:
• Inverse:
• Contrapositive:
• Biconditional(?):
Olympians are athletes
• If-Then Statement
• Converse
• Inverse
• Contrapositive
• Biconditional
All girls have brown hair
• If-Then Statement
If someone is a girl, then they have brown hair (F)
• Converse
If someone has brown hair, then they are a girl (F)
• Inverse
If someone is NOT a girl, then they don’t have
brown hair (F)
• Contrapositive
If someone does NOT have brown hair, then they
are NOT a girl (F)
• Biconditional
All triangles have 3 sides
• If-Then Statement
If an object is a triangle, then it has 3 sides.
• Converse
If an object has 3 sides, then it is a triangle.
• Inverse
If an object is NOT a triangle, then it does NOT have 3
sides
• Contrapositive
If an object does not have 3 sides then it is not a triangle.
• Biconditional
An object is a triangle IF AND ONLY IF it has 3 sides