CONDITIONAL STATEMENTS
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.