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1. What is the negation of the following: "If this triangle has two 45 degree angles then it is a right triangle."

A. this triangle does not have two 45 degree angles and it is a right triangle.
B. this triangle has two 45 degree angles or it is not a right triangle.
C. this triangle has two 45 degree angles and it is not a right triangle.
D. this triangle does not have two 45 degree angles or it is a right triangle.
E. this triangle has two 45 degree angles or it is a right triangle.
F. If this triangle does not have two 45 degree angles then it is a right triangle.
G. this triangle has two 45 degree angles and it is a right triangle.
H. If it is not a right triangle then this triangle does not have two 45 degree angles.
I. If this triangle does not have two 45 degree angles then it is not a right triangle.
J. If this triangle has two 45 degree angles then it is a right triangle.
K. If it is a right triangle then this triangle has two 45 degree angles.

Respuesta :

Answer:

Let  

  • p : this triangle has two 45 degree angles  
  • q : it is a right triangle

Step-by-step explanation:

Denote

  • r : if this triangle has two 45 degree angles then it is a right triangle

Then, r can be written  as follows

  • [tex]p \rightarrow q[/tex]

Since [tex](p \rightarrow q) \iff (\neg p \lor q)[/tex] is a tautology, you get

  • [tex] \neg(p\rightarrow q)\iff\neg(\neg p \lor q)[/tex]  

and given that  

  • [tex] \neg(p\rightarrow q)\iff(p \land \neg q) (\mbox{double negation and Morgan's Lows})[/tex]

it follows that the negation of r is

  • " this triangle has two 45 degree angles and it is not a right triangle".

Then, the correct answer is C.

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