To construct a bijection from T to R , start with the tangent function tan( x ), which is a bijection from (−π/2, π/2) to R (see the figure shown on the right). Next observe that the linear function h ( x ) = π x – π/2 is a bijection from (0, 1) to (−π/2, π/2) (see the figure shown on the left). The composite function tan( h ( x )) = tan(π x – π/2) is a bijection from (0, 1) to R . Composing this function with g ( t ) produces the function tan( h ( g ( t ))) = tan(π g ( t ) – π/2) , which is a bijection from T to R .
Three thousand years ago a man in Israel wrote: “The heavens declare the glory of God; the skies proclaim the work of his hands.” That man didn’t have a telescope or a Brittanica, but he saw something many of us today still do not see. He saw a God behind it all. It may be true that seeing a Grand Designer behind these breath-taking events requires what we call a leap of faith; but it may also be true that seeing mere coincidence behind them requires an even greater leap of faith. In my mind, much greater. But on Monday, you may be the judge.
In Logic types of fallacy are firmly described thus: First the premises and the conclusion must be statements, capable of being true or false. Secondly it must be asserted that the conclusion follows from the premises. In English the words therefore , so , because and hence typically separate the premises from the conclusion of an argument, but this is not necessarily so. Thus: Socrates is a man, all men are mortal therefore Socrates is mortal is clearly an argument (a valid one at that), because it is clear it is asserted that Socrates is mortal follows from the preceding statements. However I was thirsty and therefore I drank is NOT an argument, despite its appearance. It is not being claimed that I drank is logically entailed by I was thirsty . The therefore in this sentence indicates for that reason not it follows that .