The student is able to (I can): Apply the Law of Detachment and the Law of Syllogism in logical reasoning Write and analyze biconditional statements.

1. Obj. 10 Deductive Reasoning
Objectives
The student is able to (I can):
• Apply the Law of Detachment and the Law of Syllogism in
logical reasoning
• Write and analyze biconditional statements.

2. Recall from Inductive Reasoning:
• One counterexample is enough to
disprove a conjecture.
• If we can’t come up with a
counterexample, how can we prove that a
conjecture is true for every case?

3. deductive
reasoning
The process of using logic to draw
conclusions from given facts, definitions,
and properties.
Inductive reasoning uses specific cases and
observations to form conclusions about
general ones (circumstantial evidence).
Deductive reasoning uses facts about
general cases to form conclusions about
specific cases (direct evidence).

4. Example Decide whether each conclusion uses
inductive or deductive reasoning.
1. Police arrest a person for robbery when
they find him in possession of stolen
merchandise.
Inductive reasoningInductive reasoningInductive reasoningInductive reasoning
2. Gunpowder residue tests show that a
suspect had fired a gun recently.
Deductive reasoningDeductive reasoningDeductive reasoningDeductive reasoning

5. Most of our conjectures can be phrased as
“if p then q.” This is often written p → q.
Law of Detachment
• If p → q is a true statement and p is
true, then q is true.

6. Examples Determine if the conjecture is valid by the
Law of Detachment.
Given: If a student passes his classes, the
student is eligible to play sports.
Ramon passed his classes.
Conjecture: Ramon is eligible to play
sports.
Given: If you are tardy 3 times, you must
go to detention. Shea is in
detention.
Conjecture: Shea was tardy at least 3
times.
validvalidvalidvalid
not validnot validnot validnot valid

7. Examples
Law of Syllogism
• If p → q and q → r are true statements,
then p → r is a true statement.
Determine if each conjecture is valid by the
Law of Syllogism.
Given: If a number is divisible by 4, then it
is divisible by 2. If a number is even,
then it is divisible by 2.
Conjecture: If a number is divisible by 4,
then it is even.
x: A number is divisible by 4
y: A number is divisible by 2
z: A number is even
x → y and z → y; therefore, x → z
not validnot validnot validnot valid

8. Determine if each conjecture is valid by the
Law of Syllogism.
Given: If an animal is a mammal, then it
has hair. If an animal is a dog, then
it is a mammal.
Conjecture: If an animal is a dog, then it
has hair.
x: An animal is a mammal
y: It has hair
z: An animal is a dog
x → y and z → x, therefore z → y
or z → x and, x → y therefore z → y
validvalidvalidvalid

9. biconditional
statement
A statement whose conditional and
converse are both true. It is written as
“p if and only if qp if and only if qp if and only if qp if and only if q”, “p iff qp iff qp iff qp iff q”, or “pppp ↔↔↔↔ qqqq”.
This means that p → q is true, and q → p
is true.

10. To write the conditional statement and
converse within the biconditional, first
identify the hypothesis and conclusion,
then write p → q and q → p.
Example:
Two lines are parallel if and only if they
never intersect.
Conditional: If two lines are parallel, then
they never intersect.
Converse: If two lines never intersect, then
they are parallel.

11. Example Write the conditional and converse from the
biconditional statement.
A solution is a base iff it has a pH greater
than 7.
Conditional: If a solution is a base, then it
has a pH greater than 7.
Converse: If a solution has a pH greater
than 7, then it is a base.

12. Example
Writing a biconditional statement:
1. Identify the hypothesis and conclusion.
2. Write the hypothesis, “if and only if”,
and the conclusion.
Write the converse and biconditional from:
If 4x + 3 = 11, then x = 2.
Converse: If x = 2, then 4x + 3 = 11.
Biconditional: 4x + 3 = 11 iff x = 2.

13. Remember, for a biconditional to be true,
both the conditional and the converse must
be true.
Determine if the biconditional is true, or if
false, give a counterexample.
A quadrilateral is a square if and only if it
has four right angles.
Conditional: If a quadrilateral is a square,
then it has four right angles. TRUETRUETRUETRUE
Converse: If a quadrilateral has four right
angles, then it is a square. FALSEFALSEFALSEFALSE
(it could be a rectangle)

14. Any definition in geometry can be written
as a biconditional.
Write each definition as a biconditional:
1. A rectangle is a quadrilateral with four
right angles.
• A quadrilateral is a rectangle iff it
has four right angles.
2. Congruent angles are angles that have
the same measure.
• Angles are congruent angles iff they
have the same measure.