Question

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.

Answer 1

Let

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

Explanation:

Denote

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

Then, r can be written as follows

• 
  `p \rightarrow q`

Since
`(p \rightarrow q) \iff (\\eg p \lor q)` is a tautology, you get

• 
  `\\eg(p\rightarrow q)\iff\\eg(\\eg p \lor q)`

and given that

• 
  `\\eg(p\rightarrow q)\iff(p \land \\eg q) (\mbox{double negation and Morgan's Lows})`

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.