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Law of Detachment and Law of Syllogism

- Math 2 - Lesson 4
- Mr. Lopez

Do Now

- Write in symbolic form the following converse,

inverse and contrapositive for the following

sentence. - p ?? r
- Express a sentence logically equivalent to the

following sentence. - r ?? q

Converse r ? p Inverse p ? r Contrapositive

r ? p

Contrapositive q ? r

Law of Detachment

- If a conditional sentence is true and if the

hypothesis of the conditional is also true then,

the conclusion must also be true. - If p ? q is true, and p is true, then q must also

be true.

Using the Law of Detachment

- Based on the two given statements, what can be

concluded by using the Law of Detachment? - If M is the midpoint of a segment, then it

divides the segment into two congruent segments.

(True) - M is midpoint of AB. (True)
- By using the Law of Detachment, we can conclude

that M divides AB into two congruent segments of

AM MB

Using the Law of Detachment

- However, not all sentences can be determined with

the Law of Detachment. - Example 1
- After using the Law of Detachment is the

conclusion true? - Given
- If it is snowing, then the temperature is less

than or equal to 32 degrees. (True) - The temperature is 20 degrees. (True)
- Conclusion
- It must be snowing.
- THIS IS NOT ALWAYS TRUE. IT CAN BE LESS THAN 20

DEGREES AND NOT SNOW!

Using the Law of Detachment

- Example 2
- Given
- If a road is icy, then the driving conditions

are hazardous. (True) - Driving conditions are hazardous. (True)
- Conclusion
- NO CONCLUSION CAN BE MADE BECAUSE WE DONT KNOW

IF THE HYPOTHESIS (ICY ROAD) IS TRUE OF FALSE.

Law of Syllogism

- If p ? q is true and
- q ? r is true then
- p ? r is true.
- This law is similar to the transitive property in

algebra - If a b and b c then a c

Using the Law of Syllogism

- Example
- Given
- If a number is prime then it doesnt have

repeated factors. (True) - If a number doesnt have repeated factors, then

it is not a perfect square. (True) - According to the Law of Syllogism, we can

conclude that - If a number is prime, then it is not a perfect

square.

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