MHB G o F = {(1,3)(2,2)(3,2)(4,2)(5,5)(1,1)(2,3)(3,4)(4,5)(5,2)}

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The composition of functions F and G, denoted as F o G, is defined by applying G first and then F to the results. The correct result for F o G is {(1, 3), (2, 2), (3, 2), (4, 5), (5, 2)}, not the initial set proposed. Each output is derived from the sequential application of the functions to their respective inputs. Understanding this composition is crucial for further discussions, such as finding G o F. The thread emphasizes the importance of grasping function composition in relation to inverse functions.
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So I have the following:

F = {(1,3)(2,2)(3,2)(4,2)(5,5)}
G = {(1,1)(2,3)(3,4)(4,5)(5,2)}

Am I right in saying that F o G would be:

F o G = {(1,3)(2,2)(3,2)(4,2)(5,5)(1,1)(2,3)(3,4)(4,5)(5,2)}

If not, does F o G actually mean?

Thank you.
 
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JProgrammer said:
So I have the following:

F = {(1,3)(2,2)(3,2)(4,2)(5,5)}
G = {(1,1)(2,3)(3,4)(4,5)(5,2)}

Am I right in saying that F o G would be:

F o G = {(1,3)(2,2)(3,2)(4,2)(5,5)(1,1)(2,3)(3,4)(4,5)(5,2)}

If not, does F o G actually mean?

Thank you.
No that is not right. Surely there was a definition of the "composition" of two functions where you first met this concept? (You had, I believe, earlier posted this same "F" asking about F^{-1}. How could you possibly be dealing with inverse functions without knowing what "composition" is? The inverse function is defined by 'F o F^{-1}(x)= F^{-1}o F(x)= x for all x'.)

In any case, we can interpret "F = {(1,3)(2,2)(3,2)(4,2)(5,5)}" as meaning that F(1)= 3, F(2)= 2, F(3)= 2, F(4)= 2, and F(5)= 5. G= {(1,1)(2,3)(3,4)(4,5)(5,2)} can be interpreted a meaning that G(1)= 1, G(2)= 3, G(3)= 4, G(4)= 5, and G(5)= 2.

"F o G" means "to each x, first apply G, then apply F to that". Starting with x= 1, G(1)= 1 and F(1)= 3 so F o G(1)= 3. G(2)= 3 and F(3)= 2 so F o G(2)= 2. G(3)= 4 and F(4)= 2 so F o G(3)= 2. G(4)= 5 and F(5)= 5 so F o G(4)= 5. G(5)= 2 and F(2)= 2 so F o G(5)= 2. Written as a set of pairs, F o G= {(1, 3), (2, 2), (3, 2), (4, 5), (5, 2)}.

Now, can you use that to find G o F?
 
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I was reading a Bachelor thesis on Peano Arithmetic (PA). PA has the following axioms (not including the induction schema): $$\begin{align} & (A1) ~~~~ \forall x \neg (x + 1 = 0) \nonumber \\ & (A2) ~~~~ \forall xy (x + 1 =y + 1 \to x = y) \nonumber \\ & (A3) ~~~~ \forall x (x + 0 = x) \nonumber \\ & (A4) ~~~~ \forall xy (x + (y +1) = (x + y ) + 1) \nonumber \\ & (A5) ~~~~ \forall x (x \cdot 0 = 0) \nonumber \\ & (A6) ~~~~ \forall xy (x \cdot (y + 1) = (x \cdot y) + x) \nonumber...
I'm taking a look at intuitionistic propositional logic (IPL). Basically it exclude Double Negation Elimination (DNE) from the set of axiom schemas replacing it with Ex falso quodlibet: ⊥ → p for any proposition p (including both atomic and composite propositions). In IPL, for instance, the Law of Excluded Middle (LEM) p ∨ ¬p is no longer a theorem. My question: aside from the logic formal perspective, is IPL supposed to model/address some specific "kind of world" ? Thanks.

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