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Conditional Statements

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)

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Conditional Statements

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  1. Conditional Statements

  2. 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.

  3. 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.

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

  5. Translating English to Mathematics • English: • 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”

  6. 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

  7. 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

  8. 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

  9. Converse • The CONVERSEis 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

  10. 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

  11. 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

  12. Summing It Up • Converse • SWITCH! • Inverse • NEGATE! • Contrapositive • SWITCH AND NEGATE!

  13. 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!

  14. 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. • True! • Biconditional: An angle is right iff it measures 90. • An angle measures 90iff it is right.

  15. 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!

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