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Elementary Mathematical Logic

 How would one structure a verification of  the following?f is odd if and only if F is even and vice-versa.  F is defined as the integral of f(t) from 0 to x, dt.The key here is to know how to parse the sentence.  Let's look at this type of statement.A if and only if B.  This can be decomposed into two sentences:A if BandA only if B"A if B" can be rewritten as "If B then A" which can be written more compactly as "B-->A", or "B implies A"  We would say that B is the condition that, if met, guarantees A is a consequence.Rewriting "A only if B" is a little more tricky.  Notice that A is true only if B is true, so that if we have A, then we must also have B.  This can be rewritten as:"A only if B" means "If A then B" or "A-->B", "A implies B".It helps to think of concrete ideas.  For example, let A be "I am hungry" and B be "I will eat"Then "A-->B" means "If I am hungry, then I will eat", while"B-->A" means "If I eat, then I am hungry".One thing to notice is that the two are not equivalent.  For example, in "A-->B", A is the condition guaranteeing B is the consequence, and not the other way around, so I could still eat (B) without being hungry (A).In "B-->A", B is the condition guaranteeing A, so I could still get hungry (A) and not eat (B).Finally, the "vice-versa part" means that the pairity of f with odd and F with even must switch, so that f and even are paired, and F and odd are paired:F is odd if and only if f is even.Then a deconstruction similar to before would hold.