3. At Least One

If M is not selected then, then T must be chosen. You can then take the contrapositive: If T is not selected, then M must do it.

Now we’re going to introduce a new concept: combining conditionals. We’re just combining a statement with its contrapositive to make a new inference. In this case, the inference is “at least one.”

At The Office Video Summary

“If NOT A → B” means that at least one of A and B (or both). For example:

If Matt is not in the office tomorrow, then Tim must be there.

Diagram this as: ~M → T.

Its contrapositive is:

If Tim is not in the office tomorrow, then Matt must be there.

Diagram this as: ~T → M.

These statements mean that at least one of Matt or Tim must be in the office tomorrow, but they can also both be there.

IF NOT M →  T

Premise #1

If Matt is not in the office (M), then Tim is (T).

~M → T

Premise #2

If Tim is not in the office (T), then Matt is (M).

~T → M

Note: This is the contrapositive of premise 1.

Conclusion

At least Tim (T) or Matt must be at the office (M).

IF NOT T → M

Note: There’s nothing ruling out T and M being together.

Premise #1

If you will not attend the school trip (AST), then you must write a book report (WBR).

~AST → WBR

Premise #2

If you will not write a book report (WBR), then you must attend the school trip (AST).

~WBR → AST

Note: This is the contrapositive of premise 1.

Conclusion

You must either attend the school trip (AST) or write a book report (WBR).

IF NOT AST → WBR

Note: You can do both the AST and the WBR.

Next LSAT: January 13th

Premise #1

If you did not apply to a graduate program at the University of Oxford (O), then you should apply at the University of Cambridge (C).

~O → C

Premise #2

Contrapositive of #1:
If you did not apply for a graduate program at the University of Cambridge (C), then you should apply at the University of Oxford (O).

~C → O

Conclusion

You should apply for a graduate program at least at the University of Oxford (O) or the University of Cambridge (C).

IF NOT O C

Premise #1

If I will not buy this red shirt (R), then I will buy this blue shirt (B).

~R → B

Premise #2

Contrapositive of #1:
If I will not but this blue shirt (B), then I will buy this red shirt (R).

~B → R

Conclusion

I will buy at least the red shirt (R) or the blue shirt (B).

IF NOT R B

Premise #1

If she cannot go to work today (W), then she must attend tomorrow’s meeting (M).

~W → M

Premise #2

Contrapositive of #1:
If she cannot attend tomorrow’s meeting (M), then she must go to work today (W).

~M → W

Conclusion

She must at least go to work today (W) or attend tomorrow’s meeting (M).

IF NOT W M

Truth Tables

Negation Truth Table

P~P
 TrueFalse
FalseTrue

Conjunction ``And`` Truth Table

PQP and Q
TrueTrueTrue
TrueFalseFalse
FalseTrueFalse
FalseFalseFalse

Conjunction ``Or`` Truth Table

PQP or Q
TrueTrueTrue
TrueFalseTrue
FalseTrueTrue
FalseFalseFalse

“If NOT A → B” means that at least one of A and B (or both).

For example: If Matt is not in the office tomorrow, then Tim must be there.
Diagram: ~M → T.

Its contrapositive is: If Tim is not in the office tomorrow, then Matt must be there.
Diagram: ~T → M.

These statements mean that at least one of Matt or Tim must be in the office tomorrow, but they can also both be there.

IF NOT M →  T

Table Guide

Statement

Symbols

Description

1. If Matt is not in the office tomorrow, then Tim must be there.

~M → T

Given

2. If Tim is not in the office tomorrow, then Matt must be there.

~T → M

Given

3. At least one of Matt or Tim must be in the office tomorrow.

IF NOT M → T

Valid, Combine the 2 premises

Now that we learned how to combine statements, we can do more complex deductions, such as If and Only If.

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