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### 2.3 Deductive Reasoning

From conclusions by applying the laws of logic

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?

The inverse would be ~p⟶~q

If the value of x is not – 4,

then the square of x is not 16.

Is this True? Which statements are true?

The inverse would be ~p⟶~q

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

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.

Law of Detachment

If p ⟶q is a true conditional,

then p is true and q is true.

Does p always have to be TRUE!

Law of Detachment

If p ⟶q is a true conditional,

then p is true and q is true.

Does p always have to be TRUE!

Yes

Law of Syllogism

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.

Lets make some correct conclusion.

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

More Facts

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.

One more

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

What Conclusion can you make?

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