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cris(c)
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negating a statement...need help urgently!
Hi everyone:
I am not sure about the following thing I did. Let J be a countable finite set, and [itex]f_{jk}^{0}[/itex] and [itex]f_{jk}^{1}[/itex] be two continuous functions defined on [0,1]. Consider the following statement:
[itex]\forall lj\in J,\forall x\in[0,1],\: \, f_{jk}^{0}(x)\leq f_{jk}^{1}(x)[/itex]
Negating the above statement gives me:
[itex]\exists lj\in J,\exists \hat{x} \in[0,1],\: \, f_{jk}^{0}(\hat{x})> f_{jk} ^{1}(\hat{x})[/itex]
Question 1: Am I correct in the way I negate the original statement?
Question 2 (and perhaps the most important): The fact that the negation involves only one member gives the freedom to assume that every other element satisfies the properties in the original statement? i.e., can I assume, while constructing a proof, that [itex]\forall hz\in J[/itex] other than l and j, [itex]f_{lz}^{0}(x)\leq f_{lz}^{1}(x)\: \forall x\in[0,1][/itex]?
Thanks a lot! I truly appreciate any help you can give me!
Hi everyone:
I am not sure about the following thing I did. Let J be a countable finite set, and [itex]f_{jk}^{0}[/itex] and [itex]f_{jk}^{1}[/itex] be two continuous functions defined on [0,1]. Consider the following statement:
[itex]\forall lj\in J,\forall x\in[0,1],\: \, f_{jk}^{0}(x)\leq f_{jk}^{1}(x)[/itex]
Negating the above statement gives me:
[itex]\exists lj\in J,\exists \hat{x} \in[0,1],\: \, f_{jk}^{0}(\hat{x})> f_{jk} ^{1}(\hat{x})[/itex]
Question 1: Am I correct in the way I negate the original statement?
Question 2 (and perhaps the most important): The fact that the negation involves only one member gives the freedom to assume that every other element satisfies the properties in the original statement? i.e., can I assume, while constructing a proof, that [itex]\forall hz\in J[/itex] other than l and j, [itex]f_{lz}^{0}(x)\leq f_{lz}^{1}(x)\: \forall x\in[0,1][/itex]?
Thanks a lot! I truly appreciate any help you can give me!