CONDITIONALSTATEMENTS

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.

CONDITIONAL STATEMENT

IF A, then B.A B

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

Hypothesis: You buy a lipstick in the right place.

Conclusion: It is OK to buy the wrong lipstick.

NEGATION

The negation of A is “not A”.~A means “not A”.

S: It is raining today.~S: It is not raining today.

TRUTH VALUE

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

TRUTH VALUE

A: 2011 is the year of the rabbit. Truth value: TRUE

B: Water is solid.Truth value: False

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)

DERIVED STATEMENTS

CONDITIONAL INVERSE A B ~A ~B

CONVERSE CONTRAPOSITIVEB A ~B ~A

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)

BICONDITIONAL

CONDITIONALA B (TRUE)CONVERSEB A (TRUE)

BICONDITIONALA <--> B

BICONDITIONAL

BICONDITIONALA <--> B

A if and only if B.

DEDUCTIVE REASONING

If p q is true and p is true, then q is also true.

[(pq) ^ p] q

If p q and q r are true, then p r is also true.

[(pq) ^ (qr)] (pr)

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

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

PROVING

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 GeometryINFORMAL – essay form of a proof; spontaneous and descriptive/narrativeFORMAL – organized and well-structured

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

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