Propositions: True or False
āA statement and its contrapositive travel together ā the converse does not.ā
The formula
(p ā q) ā” (~q ā ~p)How to read it: 'if p then q' and its contrapositive 'if not q then not p' always share the same truth value
- p ā q
- ā 'if p then q' ā a conditional (the original statement)
- ~p
- ā the negation of p ā 'not p'
- ā”
- ā the two statements always have the same truth value
The hook
In math a 'proposition' isn't a feeling or an opinion ā only a sentence that comes out definitively true or false counts.
In plain words
A proposition is a sentence you can clearly judge true or false. A conditional like 'if p then q' always shares the same truth value as its contrapositive.
The intuition
If 'if it rains, the ground gets wet' is true, then the flipped 'if the ground isn't wet, it didn't rain' must be true too. It's the same fact told from the other side, so their truth values travel together.
How it's built
A proposition takes exactly two values, true or false. Negation (~) flips that value. In the conditional 'pāq', negating both p and q and reversing the direction gives the contrapositive '~qā~p', and the original and its contrapositive always share the same truth value.
Example
'if x=2 then xĀ²=4' is true. Its contrapositive 'if xĀ²ā 4 then xā 2' is also true. But the converse 'if xĀ²=4 then x=2' is false, since x=ā2 is a counterexample.
Common mistake
A true statement doesn't make its converse true. 'x=2 ā xĀ²=4' is true, but the converse 'xĀ²=4 ā x=2' is false (x=ā2). What shares the truth is the contrapositive, not the converse.
Where it's used
Every proof and logical argument in math, the conditions in a program (if), the reading of legal and contract clauses ā anywhere you weigh 'true or false'.
Where it came from
Logic that treats propositions rigorously as true-or-false objects traces back to Aristotle, and in the 1800s Boole recast it into a form you can calculate with, like algebra.
Prerequisites
Quick check
Which of the following is a 'proposition'?
- 3 is an even numberā
- this painting is beautiful
- close the window
- x+1
Practice
Which of the following is a 'proposition' ā something you can judge true or false?
Answer: 0
- A proposition = a sentence with a definite truth value
- '7 is a prime number' is a true proposition
- Opinions, questions, and single words can't be judged true or false, so they aren't propositions
Takeaway: A proposition must be judgeable as true/false ā opinions, commands, and questions are out.
Is the proposition '5 is greater than 3' true or false? Then write its negation on your notepad.
Answer: undefined
- '5 is greater than 3' is true
- Negation: '5 is not greater than 3' ā i.e. '5 ā¤ 3', which is false
- Negation flips the truth value of the original
Takeaway: Negation flips true to false and false to true.
The statement 'if it is equilateral, then it is isosceles' is true. Which of the following must also be true?
Answer: 1
- If the original is true, its contrapositive must be true too
- Contrapositive: 'if it is not isosceles, then it is not equilateral' ā true
- The converse 'if isosceles then equilateral' can be false
Takeaway: Only the contrapositive is guaranteed to share the truth ā the converse must be checked separately.
Is 'if xĀ²=9 then x=3' true? If not, jot down a counterexample on your notepad.
Answer: undefined
- If xĀ²=9 then x=3 or x=ā3
- x=ā3 gives xĀ²=9 but xā 3 ā a counterexample exists
- So the statement is false
Takeaway: A single counterexample is enough to bring a proposition down.