Understanding Conditional Statements and Logical Operators in Mathematics
This lecture covers key concepts in logic, focusing on conditional statements, their components, and various logical operations. A conditional statement takes the form p → q, expressing that if hypothesis p is true, then conclusion q follows. We explore the truth table for conditionals, examine contrapositive, converse, and inverse statements, and introduce biconditional statements. Additionally, we discuss exclusive-or, NAND, and NOR operators, illustrating their logical equivalences and applications in mathematical reasoning.
Understanding Conditional Statements and Logical Operators in Mathematics
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Presentation Transcript
Conditional Statements Lecture 2 Section 1.2 Fri, Jan 20, 2006
The Conditional • A conditional statement is a statement of the form p q • p is the hypothesis. • q is the conclusion. • Read p q as “p implies q.” • The idea is that the truth of p implies the truth of q.
Truth Table for the Conditional • p q is true if p is false or q is true. • p q is false if p is true and q is false.
Example: Conditional Statements • “If it is raining, then I am carrying an umbrella.” • This statement is true • when I am carrying an umbrella (whether or not it is raining), and • when it is not raining (whether or not I am carrying an umbrella). • It is false only if it is raining and I am not carrying an umbrella.
The Contrapositive • The contrapositive of pq is qp. • The statements pqandqp are logically equivalent.
converses qp pq inverses inverses contra positives converses pq qp The Converse and the Inverse • The converse of pq is qp. • The inverse of pq is pq.
The Biconditional • The statement p qis the biconditionalof p and q. • p q is logically equivalent to (p q) (q p).
Exclusive-Or • The statement p q is the exclusive-or of p and q. • p q is defined by
Exclusive-Or • p q means “one or the other, but not both.” • p q is logically equivalent to (pq) (qp) • p q is also logically equivalent to (p q) • p q is also logically equivalent to (pq) (qp)
The NAND Operator • The statement p| q means not bothpandq. • The operator | is also called the Scheffer stroke or NAND. • NAND stands for “Not AND.” • p | q is logically equivalent to (p q).
The NAND Operator • p| q is defined by
The NAND Operator • The three basic operators may be defined in terms of NAND. • p p | p. • p q (p | q) | (p | q). • p q (p | p) | (q | q).
The NOR Operator • The statement p q means neitherpnorq. • The operator is also called the Pierce arrow or NOR. • NOR stands for “Not OR.” • p q is logically equivalent to (p q).
The NOR Operator • p q is defined by
The NOR Operator • The three basic operators may be defined in terms of NOR. • p p p. • p q (p q) (p q). • p q (p p) (q q).
