Conditional Statements and Logic

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Conditional Statements and Logic. 2.2 Ms. Verdino. Conditional Statement. A conditional statement is a statement that can be written in if-then form. “ If _____________, then ______________.”. If the batteries are dead , then then the TV remote won't work .
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Conditional Statements and Logic2.2Ms. VerdinoConditional StatementA conditional statement is a statement that canbe written in if-then form.“If _____________, then ______________.”Ifthe batteries are dead,then then the TV remote won't work.Conditional Statements have two parts:The hypothesis is the part of a conditional statement that follows “if” (when written in if-then form.)The hypothesis is the given information, or the condition.The conclusionis the part of an if-then statement that follows “then” (when written in if-then form.) The conclusion is the result of the given information.Symbolic LogicSymbols can be used to modify or connect statements.Symbols for Hypothesis and Conclusion:Hypothesis is represented by “p”.Conclusion is represented by “q”.if p, then q or p implies qSymbolic Logicif p, then q or p implies qp  qis used to representExample:p: a number is prime q: a number has exactly two divisorspq:If a number is prime, then it has exactly two divisors.Writing A Conditional StatementConditional statements can be written in “if-then” form to emphasize which part is the hypothesis and which is the conclusion.Hint: Turn the subject into the hypothesis.Example 1:Vertical angles are congruent.can be written as...Conditional Statement:If two angles are vertical, then they are congruent.Example 2:can be written as...Seals swim.Conditional Statement:If an animal is a seal, then it swims.If……. Then vs. ImpliesAnother way of writing an if-then statement is using the word implies.If two angles are vertical, then they are congruent.Two angles are verticalimplies they are congruent.Conditional statements can be true or false.
  • A conditional statement is false only when the hypothesis is true, but the conclusion is false.
  • The truth value of a conditional statement is either true or false.
  • Statement:If you live in Tennessee, then you live in Nashville.Yes !!!Is there a counterexample?Counterexample:I live in Tennessee, BUT I live in Morristown.Therefore () the statement is false.Is the conditional true or false?If a month has 28 days, then it is February.If two angles from a linear pair, then they are supplementary.If a number is divisible by 5, then it is odd. Symbolic Logic~is used to represent the word“not”Example 1:p: the angle is obtuse~p:The angle is not obtuseNote:~p means that the angle could be acute, right, or straight.Example 2:p: I am not happy~p:I am happy~p took the “not” out- it would have been a double negative (not not)Symbolic Logicis used to represent the word“therefore”Example:Therefore, the statement is false. the statement is falseForms of Conditional StatementsConverse: Switch the hypothesis and conclusion (q  p)pqIftwo angles are vertical, thenthey are congruent.qpIftwo angles are congruent, thenthey are vertical.Forms of Conditional StatementsInverse:State the opposite of both the hypothesis andconclusion. (~p~q)pq :Iftwo angles are vertical, thenthey are congruent.~p~q:Iftwo angles are not vertical, thenthey are not congruent.Forms of Conditional StatementsContrapositive: Switch the hypothesis and conclusion and state their opposites.(~ q~p)pq : Iftwo angles are vertical, thenthey are congruent.~q~p:Iftwo angles are not congruent, thenthey are not vertical.Forms of Conditional Statements
  • Contrapositives are logically equivalent to the original conditional statement.
  • If pq is true, then qp is true.
  • If pq is false, then qp is false.
  • Compound statementA compound statement combines two or more statements. Symbolic Logic“or”is used to represent the wordp: a number is even q: a number is divisible by 3Example:pq:A number is even or it is divisible by 3.i.e.2,3,4,6,8,9,10,12,14,15,...Symbolic Logicis used to represent the word“and”p:a number is even q: a number is divisible by 3Example:A number is even and it is divisible by 3.i.e. 6,12,18,24,30,36,42...pq:Constructing a compound statements: We will go to the beachj: We will go out to dinnert: We will go to the moviessjsj s(jt) (sj) t Truth tablesCreating a truth tableBiconditional Statements
  • When a conditional statement and its converse are both true, the two statements may be combined.
  • Use the phrase if and only if (sometimes abbreviated: iff)
  • Statement: If an angle is right then it has a measure of 90.Converse:If an angle measures 90, then it is a right angle.Biconditional:An angle is right if and only if it measures 90.
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