Lessons 4–5 · 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
4
Implication, converse, inverse and contrapositive
Textbook, Part 1: pp. 23–28
GoalKnow the truth tables of implication and equivalence; form the converse, inverse and contrapositive and analyse their truth.
New words
implication · implikatsiyaconverse · konversiyacontrapositive · kontrapozitsiyanecessary and sufficient condition · zarur va yetarli shart
Explanation
The statement “if p, then q” is an implication, written p ⇒ q; here p is a sufficient condition and q a necessary condition. An implication is false in exactly one case: p true and q false; if p is false the implication is true whatever q is. The statement q ⇒ p is the converse, ¬p ⇒ ¬q the inverse and ¬q ⇒ ¬p the contrapositive. The tables show that the contrapositive is equivalent to the original implication and the inverse is equivalent to the converse; but an implication and its converse are not equivalent. The statement (p ⇒ q) ∧ (q ⇒ p) is the equivalence p ⇔ q: it is true only when p and q have the same value, and reads “p is necessary and sufficient for q”.
Worked examples
p: “x is divisible by 6”, q: “x is divisible by 3”. p ⇒ q is true. The converse q ⇒ p is false: for x = 9, q is true and p is false. The contrapositive “if x is not divisible by 3, it is not divisible by 6” is true. The inverse “if x is not divisible by 6, it is not divisible by 3” is false (again x = 9).
p: “7 is odd” (true), q: “7 is divisible by 2” (false). p ⇒ q is false (T ⇒ F), q ⇒ p is true (a false premise gives a true implication), p ⇔ q is false because the values differ. Likewise, with the false p: “2 + 2 = 5”, the implication “if 2 + 2 = 5 then 3 > 1” is true.
Class activity
“Reverse it” game: the teacher says an “if …, then …” sentence from life (for example “If it rains, the street gets wet”). Groups write the converse, inverse and contrapositive and justify which one is always true.
Practice
1
When is the implication p ⇒ q false?
Only when p is true and q is false.
2
“If a quadrilateral is a square, it is a rectangle.” Write the converse and decide whether it is true.
Converse: “If a quadrilateral is a rectangle, it is a square” — false (for example a rectangle with sides 2 and 3).
3
In how many of the 4 rows of the truth table is p ⇒ q true?
3
4
Why is the contrapositive ¬q ⇒ ¬p equivalent to p ⇒ q?
¬q ⇒ ¬p is false in just one case as well: ¬q true and ¬p false, i.e. q false and p true — exactly the row where p ⇒ q is false.