Let's consider the statement [math]p:x=3[/math], and the statement [math]q:x+1=4[/math].[br]We say that [math]p[/math] [i]implies [/i][math]q[/math], and we write [math]p\Longrightarrow q[/math] to mean that if [math]p[/math] is true, then also [math]q[/math] is true. [br](If [math]p[/math]is true, [math]q[/math] is [i]necessarily [/i]true). [br][br]The symbol [math]\Longrightarrow[/math] [i]connects [/i]a premise [math]\left(p\right)[/math] and a conclusion [math]\left(q\right)[/math] and is very used in proofs, because it's a symbolic way to show deductive reasoning.[br][br]The statement "[math]p[/math] implies [math]q[/math]" is also written "if [math]p[/math] then [math]q[/math]" or sometimes "[math]q[/math] if [math]p[/math]".[br][br]Does this sound complicated? No... let's see a few examples of implications.[br][list][*][i]If [/i]you score 68% or more in this problem, [i]then [/i]you will pass the exam.[/*][*]Your head will hurt [i]if[/i] you bang it against a wall.[/*][/list]
Given a quadrilateral [i]Q[/i], use the applet below to find out the reciprocal implications between the following statements:[br][i]a[/i]: [i]Q[/i] has an obtuse angle.[br][i]b[/i]: [i]Q[/i] has three acute angles.[br][i]c[/i]: [i]Q[/i] has no right angles. [br][br](drag the orange points to explore different quadrilaterals)
Which are the implications between statements [i]a[/i], [i]b[/i] and [i]c[/i]?
[math]b\Longrightarrow a[/math] and [math]b\Longrightarrow c[/math] because the sum of the interior angles of a quadrilateral is 360°.[br][math]c\Longrightarrow a[/math] because the interior angles cannot be all acute.
Consider this example: [math]10=11\Longrightarrow10\cdot0=11\cdot0[/math][br]We started with a false premise and implied a true conclusion.[br][br]Now consider this: [math]10=11\Longrightarrow10-1=11-1[/math][br]We started - again - with a false premise, and implied a wrong conclusion.[br][br]Implication doesn't like false premises. If we start with a false premise, the conclusion obtained by implication can be anything.
In the example above, we had the following three statements about a quadrilateral [i]Q[/i]:[br][i]a[/i]: [i]Q[/i] has an obtuse angle.[br][i]b[/i]: [i]Q[/i] has three acute angles.[br][i]c[/i]: [i]Q[/i] has no right angles. [br][br]We can say that:[br][list][*][i]a[/i] doesn't imply [i]b [/i]because a rhombus (that is not a square) has an obtuse angle, but not 3 acute ones.[/*][*][i]a[/i] doesn't imply [i]c[/i] because a right trapezoid (that is not a rectangle) contains an obtuse angle, and two right angles.[/*][*][i]c[/i] doesn't imply [i]b[/i] because a rhombus (that is not a square) has no right angles, but doesn't have three acute angles.[/*][/list][br]We explained that implication doesn't hold using a "tool" that in mathematics is named [i]counterexample[/i].[br]Click [url=https://www.geogebra.org/m/aexepwzu]here [/url]to discover more about counterexamples.