PG1.1 Statements and truth values

Simulate

Check the claim with numbers

A statement is a sentence that is either true or false, never both. "The fund beat the index in 2024 and in 2025" is one. This lesson asks what it takes for such a statement to be false.

Write $P$ for "the fund beat the index in 2024" and $Q$ for "the fund beat the index in 2025". Each is true or false, so there are four cases. The cell lists them, with two guesses at what "not ( $P$ and $Q$ )" means.

from itertools import product print("P Q not(P and Q) notP or notQ notP and notQ") for P, Q in product([True, False], repeat=2): left = not (P and Q) guess_or = (not P) or (not Q) guess_and = (not P) and (not Q) print(f"{P!s:6} {Q!s:6} {left!s:13} {guess_or!s:13} {guess_and!s}")

⌘/Ctrl + Enter runs the cell

The "or" guess matches the left column in all four rows; the "and" guess fails in rows 2 and 3. In row 2 the fund beat the index in 2024 but not in 2025. The statement is false, yet "it lost in both years" is false too.

The next cell counts the rows where two statements agree, for three candidate laws.

from itertools import product rows = list(product([True, False], repeat=2)) laws = { "not(P and Q) = notP or notQ": lambda P, Q: (not (P and Q)) == ((not P) or (not Q)), "not(P or Q) = notP and notQ": lambda P, Q: (not (P or Q)) == ((not P) and (not Q)), "not(P and Q) = notP and notQ": lambda P, Q: (not (P and Q)) == ((not P) and (not Q)), } for name, same in laws.items(): agree = sum(same(P, Q) for P, Q in rows) print(f"{name}: agree in {agree} of 4 rows")

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The first two laws agree in all 4 rows. The third agrees in only 2, so it is not a law.

Conjecture

P

Two statements have only four cases between them, and the table checked all four. So here the table is a complete check, a proof by exhaustion. The proof below gives the reason in words, the kind of argument that still works later, when the cases are too many to list.

Statements and how they combine

Lesson 0.2.1 met statements that are true or false about a set. This section gives the rules for building new statements from old ones.

A question ("Is the index up?") or a command ("Buy the index") is not a statement. Neither is " $x > 3$ " on its own, since it is true for some $x$ and false for others. Lesson PG1.3 turns such sentences into statements.

Three words build new statements: "not", written $\neg$ ; "and", written $\wedge$ ; and "or", written $\vee$ . Each is defined by the truth values it gives:

$P$	$Q$	$\neg P$	$P \wedge Q$	$P \vee Q$
true	true	false	true	true
true	false	false	false	true
false	true	true	false	true
false	false	true	false	false

"And" is true only when both parts are true. "Or" is true when at least one part is true, both included. Everyday English sometimes means "one or the other, not both", but in mathematics "or" always allows both.

Rule: Equivalent statements

A \equiv B

Two statements $A$ and $B$ are equivalent when they have the same truth value in every case. To prove it, show that $A$ is true exactly when $B$ is.

P

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Format 1

Reorder the steps

Here is a proof of the first law in words, with its lines shuffled. Put them in order with the arrows, or drag a line by its handle.

First suppose $\neg(P \wedge Q)$ is true. Then $P \wedge Q$ is false.
So $\neg P$ is true or $\neg Q$ is true, which makes $\neg P \vee \neg Q$ true.
An 'and' is true only when both parts are true, so at least one of $P$ and $Q$ is false.
Now suppose $\neg P \vee \neg Q$ is true. Then $P$ is false or $Q$ is false.
Either way $P \wedge Q$ has a false part, so it is false, and $\neg(P \wedge Q)$ is true.
Take any two statements $P$ and $Q$ . We show that $\neg(P \wedge Q)$ and $\neg P \vee \neg Q$ are true in exactly the same cases.
Each statement is true exactly when the other is, so they are equivalent.

The second law has the same shape of proof, with "and" and "or" swapped. The table in the simulation already settles it, since it checked all four cases.

Format 3

Find the error

This proof claims that $\neg(P \wedge Q)$ is equivalent to $\neg P \wedge \neg Q$ . The simulation found that claim false in two rows, so one line must be wrong. Select it, then say what is wrong with it.

So the statement about 2024 and 2025 is false when the fund missed in 2024, in 2025, or in both. One bad year is enough.

x > 3

Statements and truth values

Check the claim with numbers

Statements and how they combine

Rule: Equivalent statements

Reorder the steps

Find the error

Summary card

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