☰ Contents · Mathematics

Statements and logical operations

Lessons 2–3 · 2 lessons · M. A. Mirzaahmedov, Sh. N. Ismailov, A. Q. Amanov (algebra and analysis), B. Q. Haydarov (geometry). Mathematics Grade 10, Parts 1 and 2, 1st edition. EXTREMUM PRESS, Tashkent, 2017
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Logical equivalence and laws of logic

Textbook, Part 1: pp. 21–23
GoalBuild the truth table of a compound statement; prove a tautology and logical equivalence with a table; use the laws for ¬(p ∧ q) and ¬(p ∨ q).
New words
truth table · rostlik jadvalitautology · tavtologiyalaw of logic · mantiqiy qonunlogically equivalent · mantiqiy tengkuchli
Explanation

We build the truth table of a compound statement step by step: first list all value pairs of p and q (TT, TF, FT, FF), then fill the column of each part, and finally the column of the whole expression. Two statements give 4 rows, three statements give 2 · 2 · 2 = 8 rows. A statement that is true in every row is a tautology or law of logic; for example p ∨ ¬p is always true. If the last columns of two tables coincide, the statements are logically equivalent, written p ≡ q (or p ⇔ q). Two key equivalences are ¬(p ∧ q) ≡ ¬p ∨ ¬q and ¬(p ∨ q) ≡ ¬p ∧ ¬q: the negation of “both” is “at least one is not”, and the negation of “at least one” is “neither”.

Worked examples
Compare ¬(p ∨ q) and ¬p ∧ ¬q. For rows (p, q) = TT, TF, FT, FF the value of p ∨ q is T, T, T, F, so ¬(p ∨ q) is F, F, F, T. For ¬p ∧ ¬q: ¬p = F, F, T, T and ¬q = F, T, F, T, and their conjunction is F, F, F, T. The columns agree, so the statements are equivalent.
The negation of “x > 2 and x < 5” is “x ≤ 2 or x ≥ 5” (¬(p ∧ q) ≡ ¬p ∨ ¬q). The negation of “x < −1 or x > 4” is “x ≥ −1 and x ≤ 4”, that is −1 ≤ x ≤ 4 (¬(p ∨ q) ≡ ¬p ∧ ¬q).
Class activity

In pairs: one student says a statement (“I am right-handed and I know English”), the other says its negation using “or”, and together they check the equivalence with a table.

Practice
1
What is a tautology? Give an example.
2
Is p ∧ ¬p a tautology? Answer by the table rows.
3
How many rows does a truth table of three statements p, q, r have?
4
Why is the negation of “both are true” equal to “at least one is false”? Explain with ¬(p ∧ q).