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 B
and
A 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.
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 B
and
A 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.