Conditional Statements

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Presentation transcript:

Conditional Statements

Conditional Statements A CONDITIONAL STATEMENT is a logical statement using the words “IF” and “THEN” Example: IF I do my chores, THEN I get my allowance.

Conditional Statements There are two parts to Conditional Statements: The HYPOTHESIS (the IF part) The CONCLUSION (the THEN part) Example: IF I do my chores, THEN I get my allowance.

Symbolic Notation Conditional Statements can be written in Symbolic Notation The HYPOTHESIS is marked by the letter p The CONCLUSION is marked by the letter q Example p: “I do my chores” q: “I get my allowance”

Translating English to Mathematics IF I do my chores, THEN I get my allowance Mathematics: Let p be “I do my chores” Let q be “I get my allowance” p q Read “p implies q”

Examples IF I come to school late, THEN I will get a tardy pass. IF I lie to my parents, THEN I’ll be grounded Notes Examples

Negation A statement can be altered by negation Doing the OPPOSITE The symbol for negation is ~ Example Statement: We are in school Negation: We are NOT in school Notes Examples

Converse, Inverse, Contrapositive Recall our original Conditional Statement If I do my chores, then I get my allowance Using this Conditional, we can write three other statements Converse Inverse Contrapositive

Converse The CONVERSE is formed by switching the hypothesis and conclusion (SWITCH) Original Conditional p q If I do my chores, then I get my allowance Converse q p If I get my allowance, then I did my chores Notes Examples

Inverse The INVERSE is formed by negating the hypothesis and the conclusion of the original statement (NEGATE) Original Conditional p q If I do my chores, then I get my allowance Inverse ~p ~q If I my DON’T do my chores, then I DON’T get my allowance Notes Examples

Contrapositive The CONTRAPOSITIVE is formed when you negate the converse (SWITCH AND NEGATE) Original Conditional p q If I do my chores, then I get my allowance Contrapositive ~q ~p If I DON’T get my allowance, then I DIDN’T do my chores Notes Examples

Summing It Up Converse Inverse Contrapositive SWITCH! NEGATE! SWITCH AND NEGATE!

BICONDITIONALS When a conditional statement and its converse are both true, the two statements can be combined. Use the phrase IF AND ONLY IF (abbreviated: IFF) Symbolic Notation p q Remember, p q AND q p BOTH must be true!

BICONDITIONAL Example Conditional: If an angle is right, then it has a measure of 90. True! Converse: If an angle has a measure of 90, then it is right. Biconditional: An angle is right iff it measures 90. An angle measures 90 iff it is right.

BICONDITIONALS NON-EXAMPLE Conditional: If we are in Geometry class, then we are in school. True! Converse: If we are in school, then we are in Geometry class. Not always true! Can’t be written as a BICONDITIONAL!