Thinking Mathematically

1
Thinking Mathematically

• Equivalent Statements, Conditional Statements,
  and De Morgans Laws

2
Equivalent Statements
Two statements are equivalent if they have the
same truth values.
3
A Conditional Statement and Its Equivalent
Contrapositive

• p ? q q ? p
• The truth value of a conditional statement does
  not change if the antecedent and consequent are
  reversed and both are negated. The statement
• q ? p is called the contrapositive of the
  conditional p ? q.

4
Converse and Inverse

• q ? p is the converse of p ? q. p ? q is the
  inverse of p ? q.

The converse and inverse are contrapositives and
are equivalent.
5
Conditional Statements
Let p and q be statements.

• Name Symbolic Form
• Conditional p ? q
• Converse q ? p
• Inverse p ? q
• Contrapositive q ? p

6
The Negation of a Conditional Statement

• The negation of p ? q is p /\ q. This can be
  expressed as
• (p ? q) p /\ q.

7
De Morgans Laws

• (p/\q) p\/q

8
De Morgans Laws

• (p\/q) p/\q

9
Thinking Mathematically

• Equivalent Statements, Conditional Statements,
  and De Morgans Laws