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
2
Statements: negation, conjunction and disjunction
Textbook, Part 1: pp. 14–20
GoalRecognise a statement; know the truth tables of negation, conjunction and disjunction and show their truth sets on Venn diagrams.
A statement is a declarative sentence that is true or false; a question, a command or a personal opinion (“this colour is nice”) is not a statement. Statements are named p, q, r, and the truth values are written T (true) and F (false). The negation ¬p (“not p”) is false when p is true and true when p is false. The conjunction p ∧ q (“p and q”) is true only when both statements are true; the disjunction p ∨ q (“p or q”) is true when at least one is true and false only when both are false. If a statement depends on a variable x, the set of x that make it true is its truth set: if P is the truth set of p, then ¬p has P′, p ∧ q has P ∩ Q and p ∨ q has P ∪ Q.
Worked examples
p: “17 is prime” (true), q: “17 is divisible by 3” (false). Then ¬p is false, ¬q is true, p ∧ q is false (the second part is false), p ∨ q is true (the first part is true).
U = {x | 1 ≤ x ≤ 12, x ∈ Z}, p: “x is a multiple of 4”, q: “x is odd”. P = {4, 8, 12}, Q = {1, 3, 5, 7, 9, 11}. P ∩ Q = ∅, so p ∧ q is true for no x; n(P ∪ Q) = 3 + 6 = 9; the truth set of ¬p has 12 − 3 = 9 elements.
Class activity
“True or false?” game: the teacher states two statements (for example “8 is even” and “8 is prime”); students call out T or F for the statement formed with “and”, “or”, “not”. Whoever errs sits out.
Practice
1
Which are statements: a) “Is 9 a square?”, b) “91 is prime”, c) “Close the door”, d) “21 + 19 = 40”? Give each statement’s value.
b) and d). b) is false (91 = 7 · 13), d) is true.
2
Write the negations of “a ≥ 6” and “triangle ABC is isosceles”.
a < 6; triangle ABC is not isosceles.
3
U = {1, 2, …, 20}, p: “x is a multiple of 3”, q: “x is prime”. Find n(P ∪ Q) (there are 6 multiples, 8 primes, and only 3 is both).
13
4
Why is p ∨ q false only when both p and q are false?
“Or” asks for at least one true statement; as soon as one is true the condition is met. It fails only when both are false.