Inverses
There are two other statements besides the converse for a conditional statement. One is the inverse and the other is the contrapositive. The inverse of a statement is the negation of the hypothesis and conclusion of a conditional statement. For example, the statement, "If it is July 4th, then it is Independence Day." has an inverse of, "If it is not July 4th, then it is not Independence Day." Given a conditional statement of \(p\rightarrow q\) the inverse is \(\sim p\rightarrow \sim q\). An inverse has the logical equivalence of the converse. This means that if the converse is true, then the inverse is true.
Contrapositive
The contrapostive of a conditional statement is the inverse of the converse. For example, the statement, "If an angle has a measure of \(40^{\circ}\), then it is acute." has a contrapositive of, "If an angle is not acute, then it does not have a measure of \(40^{\circ}\)." Given a conditional statement of \(p\rightarrow q\) the contrapositive is \(\sim q\rightarrow \sim p\). A contrapositive has the logical equivalence of the conditional statement.
Law of Syllogism
The law of syllogism allows us to combine logical or conditional statements together using transitivity (chain of reasoning). If \(p\rightarrow q\) and \(q\rightarrow r\), then \(p\rightarrow r\). An example would be, "If it is raining, then you will need to turn your windshield wipers on." "When you are using your windshield wipers, your headlights must be turned on." Using the Law of Syllogism these statements can be combined into, "If it is raining, then your headlights must be turned on."
Example 1:
Use the Law of Syllogism to to write the conclusion that can be reached using all of the given statements.
If I reach my potential, then my learning does not suffer.
If I am prepared for class, then I finished my homework.
If I watch TV all evening, then I will not finish my homework.
If I am not prepared for class, then my learning will suffer.
Solution:
Let's represent the statements using logic shorthand.
If I reach my potential, then my learning does not suffer. \(p\rightarrow \sim q\)
If I am prepared for class, then I finished my homework. \(r\rightarrow s\)
If I watch TV all evening, then I will not finish my homework. \(t\rightarrow \sim s\)
If I am not prepared for class, then my learning will suffer. \(\sim r\rightarrow q\)
We can rearrange and rewrite these statements into a chain of reasoning and contrapositives. We look for the two phrases that not used twice: \(p\) and \(t\).
\(p\rightarrow \sim q\), \(\sim q\rightarrow r\), \(r\rightarrow s\), \(s\rightarrow\sim t\)
This can be combined into the conclusion:
\(p\rightarrow \sim t\) which is written as "If I reach my potential, then I do not watch TV all evening."
Try these problems.
1) Write the converse, inverse and contrapositive of:
If is not raining, then I will go to the beach.
2) Write the converse, inverse and contrapositive of:
If a quadratic function has a positive leading coefficient, then its graph will cross the \(x\)-axis.
Use the Law of Syllogism to to write the conclusion that can be reached using all of the given statements.
3) \(w\rightarrow p\)
\(r\rightarrow s\)
\(p\rightarrow t\)
\(s\rightarrow w\)
4) \(\sim e \rightarrow f\)
\(g\rightarrow\sim k\)
\(f\rightarrow g\)
\(m\rightarrow d\)
\(\sim m\rightarrow k\)
5) \(p\rightarrow\sim c\)
\(z\rightarrow a\)
\(\sim z\rightarrow\sim x\)
\(a\rightarrow p\)
\(\sim w\rightarrow c\)
6) If Catnip burns Cotta, then Dogwire drops Tiebars.
If Zippo eats Watso, then Nickel rots Lumpy.
If Google fries Bingle, then Zippo eats Watso.
If Nickel rots Lumpy, then Catnip burns Cotta.
The author Charles Dodgson aka Lewis Carroll who wrote Alice in Wonderland was also a mathematician that wrote many logic puzzles. Use the Law of Syllogism to to write the conclusion that can be reached from some of his puzzles.
7) Babies are illogical;
Nobody is despised who can manage a crocodile;
Illogical persons are despised.
8) My saucepans are the only things I have that are made of tin;
I find all your presents very useful;
None of my saucepans are of the slightest use.
9) No potatoes of mine, that are new, have been boiled;
All my potatoes in this dish are fit to eat;
No unboiled potatoes of mine are fit to eat.
10) No ducks waltz;
No officers ever decline to waltz;
All my poultry are ducks.
11) Every one who is sane can do Logic;
No lunatics are fit to serve on a jury;
None of your sons can do Logic.
12) No one takes in the Times, unless he is well-educated;
No hedge-hogs can read;
Those who cannot read are not well-educated.
13) Showy talkers think too much of themselves;
No really well-informed people are bad company;
People who think too much of themselves are not good company.
14) All hummingbirds are richly coloured;
No large birds live on honey;
Birds that do not live on honey are dull in colour.
15) Create your own syllogism.
Solution Bank