Difference between revisions of "1974 AHSME Problems/Problem 13"
m (added category) |
|||
| Line 29: | Line 29: | ||
{{AHSME box|year=1974|num-b=12|num-a=14}} | {{AHSME box|year=1974|num-b=12|num-a=14}} | ||
[[Category:Logic Problems]] | [[Category:Logic Problems]] | ||
| + | {{MAA Notice}} | ||
Revision as of 11:43, 5 July 2013
Problem
Which of the following is equivalent to "If P is true, then Q is false."?
Solution
Remember that a statement is logically equivalent to its contrapositive, which is formed by first negating the hypothesis and conclusion and then switching them. In this case, the contrapositive of "If P is true, then Q is false." is "If Q is true, then P is false." 
The fact that a statement's contrapositive is logically equivalent to it can easily be seen from a venn diagram arguement.
![[asy] draw(circle((0,0),3)); draw(circle((-1,1),1)); label("$q$",(1,-1)); label("$p$",(-1,1)); [/asy]](http://latex.artofproblemsolving.com/c/7/2/c72470077b3f2310bd9671afbb8f766bc7ca8480.png)
From this venn diagram, clearly "If 
, then 
." is true. However, since is fully contained in , the statement "If not , then not ." is also true, and so a statement and its contrapositive are equivalent.
See Also
| 1974 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 12 |
Followed by Problem 14 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
