This document defines and provides examples of conditional statements, truth values, derived statements such as converses and contrapositives, biconditionals, and deductive reasoning. Conditional statements relate an if-part (hypothesis) and then-part (conclusion). Truth values are either true or false. Derived statements like converses and contrapositives relate to the original conditional statement. Deductive reasoning uses conditional statements and logical rules to derive true conclusions.

1. CONDITIONAL
STATEMENTS

3. CONDITIONAL STATEMENT
A conditional statement is a
statement in IF and THEN form. The
IF part is called the hypothesis and
the THEN part is called the
conclusion.

4. CONDITIONAL STATEMENT
IF A, then B.
A  B

5. If you buy a lipstick in the right place,
then it’s OK to buy the wrong lipstick.

6. Hypothesis: You buy a lipstick in the
right place.
Conclusion: It is OK to buy the wrong
lipstick.

7. NEGATION
The negation of A is “not A”.
~A means “not A”.
S: It is raining today.
~S: It is not raining today.

8. TRUTH VALUE
Truth value of a statement is
either TRUE or FALSE.
(Valid vs. Invalid)

9. TRUTH VALUE
A: 2011 is the year of the
rabbit.
Truth value: TRUE
B: Water is solid.
Truth value: False

10. TRUTH VALUE
A statement and its negation
have different truth value.
B: A frog is a bird. (FALSE)
~B: A frog is not a bird. (TRUE)

12. DERIVED STATEMENTS
CONDITIONAL INVERSE
A  B ~A  ~B
CONVERSE CONTRAPOSITIVE
B  A ~B  ~A

15. THEOREM
A conditional and its
corresponding contrapositive are
logically equivalent. (Same truth
value). The converse and inverse
of a conditional are logically
equivalent. (Same truth value)

17. BICONDITIONAL
CONDITIONAL
A  B (TRUE)
CONVERSE
B  A (TRUE)
BICONDITIONAL
A <--> B

18. BICONDITIONAL
BICONDITIONAL
A <--> B
A if and only if B.

19. DEDUCTIVE
REASONING

20. If p  q is true and p is
true, then q is also true.
[(pq) ^ p]  q

21. If p  q and q  r are true,
then p  r is also true.
[(pq) ^ (qr)]  (pr)

22. All AA students are female.
“If a student is an AA student, then the student is a
female.” (TRUE)
FACT/Given: Sam is an AA student.
(TRUE)
Conclusion: Sam is female. (TRUE)
by Law of Detachment

23. All AA students are female.
“If a student is an AA student, then the student is a
female.” (TRUE)
All females have XY chromosomes.
“If you are female, then you have XY chromosomes.”
Conclusion: If a student is an AA student, then the
student has XY chromosomes.. (TRUE)
by Law of Syllogism
AA  F and F  XY
therefore AA  XY

24. PROVING

25. To prove a conjecture, we apply
deductive reasoning.
To prove something we need to supply a proof.
Truth is based on solid evidences (proofs).
A proof is a logical argument in which each
statement you make is supported by a statement
that is accepted as true
Forms of Proof in Geometry
INFORMAL – essay form of a proof; spontaneous
and descriptive/narrative
FORMAL – organized and well-structured

26. A group of algebraic steps used to solve
problems form a deductive argument.

27. A two-column proof, or formal proof, contains
statements and reasons organized in two columns.