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# 2.3 Deductive Reasoning - PowerPoint PPT Presentation

2.3 Deductive Reasoning. From conclusions by applying the laws of logic. Symbolic Notation. Conditional statement If p , then q p ⟶q Converse q⟶p Biconditional p ⟷ q. Let p be “the value of x is – 4” Let q be “ the square of x is 16”. Write p ⟶q Write q⟶p.

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### 2.3 Deductive Reasoning

From conclusions by applying the laws of logic

Conditional statement

If p, then qp⟶q

Converseq⟶p

Biconditional p ⟷ q

Let p be “the value of x is – 4”Let q be “ the square of x is 16”

Write p⟶q

Write q⟶p

Let p be “the value of x is – 4”Let q be “ the square of x is 16”

Write p⟶q

If the value of x is – 4,

then the square of x is 16

Write q⟶p

If the square of x is 16,

then the value of x is – 4

Is p⟷qTrue?

Write the Contrapositive

The negation of p “the value of x is -4” is written as ~p; meaning “the value of x is not -4”

The contrapositive ~q⟶~p

If the square of x is not 16, then the value of x is not -4.

Is this true?

If the value of x is not – 4,

then the square of x is not 16.

Is this True? Which statements are true?

If the value of x is not – 4,

then the square of x is not 16.

Is this True? Which statements are true?

The conditional statement and the contrapositive are both true.

The converse and inverse are both false.

Remember the conditional and the contrapositive are equivalent statement as are the converse and the inverse.

Deductive reasoning uses known facts, definitions and postulates to make a logical argument.

Logical arguments follow laws and methods.

Here are two laws of logical

Law of Detachment and the Law of Syllogism.

If p ⟶q is a true conditional,

then p is true and q is true.

Does p always have to be TRUE!

If p ⟶q is a true conditional,

then p is true and q is true.

Does p always have to be TRUE!

Yes

Here we have a chain is true statements linked together.

If p⟶qand q⟶r, then p⟶r

Here we have a chain is true statements linked together.

If p⟶qand q⟶r, then p⟶r.

I go to school at Marian High school.

Marian High school is in Mishawaka.

I go to school in Mishawaka.

If a fish swims at 68 mi/h, then it swims at 110 km/h.

If a fish can swim at 110 km/h, then it is a sailfish

If a fish is the largest species of fish, then it is a Great White Shark

If a fish weights over 2000 lbs, then it is the largest species of fish.

If a fish is the fastest species of fish, then it can reach speeds of 68 mi/h.

What Conclusion can you make?

Page 91 – 93

#8 – 20 even

24 – 34 even, 45 - 48

Page 91 -93

#9 – 19 odd

23 – 35 odd

36 – 42 even, 49