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 |
True | False |
False | True |
Conjunction ``And`` Truth Table
P | Q | P and Q |
True | True | True |
True | False | False |
False | True | False |
False | False | False |
Conjunction ``Or`` Truth Table
P | Q | P or Q |
True | True | True |
True | False | True |
False | True | True |
False | False | False |
“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.