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In mathematics, a relation is a set of ordered pairs, where each input is related to one or more outputs. A function is a specific type of relation where each input is related to exactly one output. Therefore, all functions are relations because they involve a set of ordered pairs, but not all relations are functions because some relations may have an input related to multiple outputs, violating the definition of a function.
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ViviI'm pretty sure that no one has actually wrote a complete proof of this statement, but the concept centers around the fact that functions are relations where the codomain is dependent on the domain, but relations don't necessarily have to be. If that sounds vague, it's because no one has really came up with a precise answer yet.
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Actually, the definition of a function ensures that it is a relation. Given that, all you need to do is to find one relation that is not a function. A popular one is y = sqrt(x) for x ≥ 0. Since each value of x (other than 0), is mapped onto 2 distinct values of y the relation is not a function. However, it is easily made into function by limiting the codomain to non-negative reals or to non-positive reals.
Similarly, relations such as reciprocal or logarithm can be made into functions by defining the domain or codomain to get around the exceptions.
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When do you say that a given function is velation?
I assume you mean "relation". By definition, all functions are relations; but only some relations are functions.
Do all functions have integrals?
No, all functions are not Riemann integrable
Are all functions continuous?
No. Not all functions are continuous. For example, the function f(x) = 1/x is undefined at x = 0.
How do you determine the domain and range of relations and functions?
Some functions are only defined for certain values of the argument. For example, the the logarithm is defined for positive values. The inverse function is defined for all non-zero numbers. Sometimes the range determines the domain. If you are restricted to the real numbers, then the domain of the square root function must be the non-negative real numbers. In this way, there are definitional domains and ranges. You can then chose any subset of the definitional domain to be your domain, and the images of all the values in the domain will be the range.
X equals x demonstrates what property of equality?
The reflexive property, which is a property of all equivalence relations. Two other properties, besides reflexivity, of equivalence relations are: symmetry and transitivity.
