Conditional Statements

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Lesson 4.1

• Conditional Statements

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Conditional Statement

• a statement with two parts, a hypothesis and a
  conclusion, also called an if-then statement
• Example Two points are collinear if they lie on
  the same line.
• if-then form If two points lie on the same line,
  then they are collinear.
• Hypothesis two points lie on the same line
• Conclusion the points are collinear

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You try it!

• Write the following conditional statement in
  if-then form.
• An angle with measure of 90? is a right angle.

• If an angle measures 90?, then it is a right
  angle

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Converse of a conditional statement

• A statement formed by switching the hypothesis
  and the conclusion.
• Example
• Conditional statement If a number is divisible
  by 3, then it is odd.
• Converse If a number is odd, then it is
  divisible by 3.

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You try it!

• First write the conditional statement as an
  if-then statement. Then write the converse.
• All monkeys have tails.

• Conditional Statement
• If an animal is a monkey, then it has a tail.
• Converse
• If an animal has a tail, then it is a monkey

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Negation

• the negative of a statement
• Examples m?A 30? negation - m?A ?
  30?
• ?A is acute negation - ?A is not acute
• You try it!
• Write the negation of the statement.
• My favorite color is green.

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Inverse

• When you negate the hypothesis and conclusion of
  a conditional statement.
• For example
• Original Conditional Statement
• if ?A 30?, then ?A is acute.
• Inverse
• if ?A ? 30?, then ?A is not acute.

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Contrapositive

• when you negate the hypothesis and conclusion of
  the converse of a conditional statement
• Example If a number is divisible by nine,
  it is divisible by 3.
• contrapositive If a number is not divisible by
  3, then it is not divisible by 9.

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You try it!

• Write the contrapositive of the conditional
  statement.
• If two planes intersect, then their intersection
  is a line.

• If the intersection of two planes is not a line,
  then the two planes do not intersect.

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Equivalent Statements

• Two statements that are either both true or both
  false.

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Biconditional Statement

• a statement that contains if and only if
  equivalent to writing a conditional statement and
  its converse.
• For a biconditional statement to be true, both
  the conditional statement and its converse must
  be true.
• Can be written forward and backward.
• All definitions can be written as a true
  biconditional statement.
• Example Three lines are coplanar if and only if
  they lie in the same plane.

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Symbolic notation to write conditional statements

• p represents the hypothesisq represents the
  conclusion
• conditional statement If p, then q can be
  written as p ? q.
• converse If q, then p can be written as q ?
  p.
• biconditional statement p if and only if q can
  be written as p ? q.

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Symbolic notation cont.

• negationNot p can be written as p.
• inverseIf not p, then not q can be written as
  p? q.
• contrapositive If not q, then not p can be
  written as q? p.

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Deductive reasoning

• uses facts, definitions, and accepted properties
  in order to write a logical argument.
• Example Josh knows that Brand X computers
  cost less than Brand Y computers. He also knows
  that Brand Y computers cost less than Brand Z.
  Josh reasons that Brand X costs less than Brand Z.

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Inductive reasoning

• uses previous examples and patterns to form a
  conjecture.
• Example Josh knows that Brand X computers
  cost less than Brand Y computers. All other
  brands of computers that Josh knows of cost less
  than Brand X. Josh reasons that Brand Y costs
  more than all other brands.

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Law of Detachment

• If p? q is a true conditional statement and p is
  true, then q is true.
• Example If two angles form a linear pair,
  then they are supplementary ?A and ?B are a
  linear pair. So, ?A and ?B are supplementary.

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Law of Syllogism

• If p? q and q? r are true conditional statements,
  then p? r is true.
• CONNECTION Deductive reasoning uses the Law of
  Syllogism to form logical arguments.