What does the symbol ≡ mean in this optimization expression?

[michonamona]

What does the symbol ≡ mean?

For example:

$$max_{x}[F(x,\alpha):g(x)\leq b,x\geq 0] \equiv G(\alpha,b)$$

Thanks for your help.

M

 

[D H]

In this case, it appears to means "defined as". It can also mean "identically equal to", as for example $\sin^2 \theta + \cos^2\theta \equiv 1$.

 

[michonamona]

In this case, it appears to means "defined as". It can also mean "identically equal to", as for example $\sin^2 \theta + \cos^2\theta \equiv 1$.

So in your trigonometric example, its like saying "it may as well be 1, but its not exactly 1". Is this what it means?

 

[D H]

No! In the trigonemtric example in means, in English, that the sum of the squares of the sine and cosine of an angle is always exactly one. It is an identity.

 

[Mark44]

No, not at all. It means "identically equal to" in this identity. This means that the left side of this identity is equal to 1 for all values of theta.

Here are a couple of examples to distinguish = from $\equiv$.

2x + 4 $\equiv$ 2(x + 2). This equation is an identity, one that is true for all values of x. You cannot solve for x in this equation.

2x + 4 = 8. This equation is a conditional equality, one that is true only for one value of x. You can solve for the value of x in this equation.