1. If – Then Statements
Lesson 1.5

2. If it rains today, then we will not go to the pool.
If p then q is called a Conditional Statement (sometimes shown as p  q or as “p implies q”)
p is the input, hypothesis, antecedent
q is the output, conclusion, consequent
Biconditional: p q means p  q and q  p

3. Truth Table for p  q ** The one to pay attention to is F  T ;
because there is no counterexample,
this statement is true.

4. Example 1 Give the truth value of the conditional.
If 4 > 9, then 4 > 7
If 6 < 8, then 6 < 10
If 5 < 3, then 5 < 9
False  False; truth value: True
True  True; truth value: True
False  True; truth value: True

5. Contrapositive, Converse, Inverse of a Statement
Given Conditional
Its Contrapositive
Its Converse
Its Inverse
p  q
~ q  ~ p
q  p
~ p  ~ q

6. Cars Conditional: If a car is a neon, then it is a Dodge.
Neons
Conditional: If a car is a neon, then it is a Dodge.
Contrapositive: If the car is not a dodge, then it is not a neon.
Converse: If the car is a dodge, then it is a neon. (false)
Inverse: If the car is not a neon, then it is not a dodge. (false)
Dodge
Cars

7. Statement: the conditional has the same truth value as its contrapositive.
Proof: P Q
P  Q ~Q ~P
~Q  ~P T F P Q
P  Q ~Q ~P
~Q  ~P

8. Try one!
Write the contrapositive, converse, and inverse of the universal conditional:
functions f, if f is a logarithm function, then f(1) = 0.
Contrapositive: functions f, If f(1) ≠ 0, then f is not a logarithm function.
Converse: functions f, If f(1) = 0, then f is a logarithm function.
Inverse: functions f, If f is not a logarithm function, then f(1) ≠ 0. A A A A

9. Homework Page 45 2-22(evens) 24: extra credit!