The (mathematical) logic behind the scenes - Lesson+Exploration+Practice
Material implication: ⇒
Let's consider the statement , and the statement .
We say that implies , and we write to mean that if is true, then also is true.
(If is true, is necessarily true).
The symbol connects a premise and a conclusion and is very used in proofs, because it's a symbolic way to show deductive reasoning.
The statement " implies " is also written "if then " or sometimes " if ".
Does this sound complicated? No... let's see a few examples of implications.
- If you score 68% or more in this problem, then you will pass the exam.
- Your head will hurt if you bang it against a wall.
Exploring implications in geometry
Given a quadrilateral Q, use the applet below to find out the reciprocal implications between the following statements:
a: Q has an obtuse angle.
b: Q has three acute angles.
c: Q has no right angles.
(drag the orange points to explore different quadrilaterals)
Which are the implications between statements a, b and c?
Implication is confused by fake guys
Consider this example:
We started with a false premise and implied a true conclusion.
Now consider this:
We started - again - with a false premise, and implied a wrong conclusion.
Implication doesn't like false premises. If we start with a false premise, the conclusion obtained by implication can be anything.
Showing why things go wrong
In the example above, we had the following three statements about a quadrilateral Q:
a: Q has an obtuse angle.
b: Q has three acute angles.
c: Q has no right angles.
We can say that:
- a doesn't imply b because a rhombus (that is not a square) has an obtuse angle, but not 3 acute ones.
- a doesn't imply c because a right trapezoid (that is not a rectangle) contains an obtuse angle, and two right angles.
- c doesn't imply b because a rhombus (that is not a square) has no right angles, but doesn't have three acute angles.