without loss of generality, we use vs respectively and as shown in figure

These two phrases are not directly comparable as they serve different purposes. "Without loss of generality, we use" is a common phrase used in mathematics and other technical fields to simplify proofs by assuming a particular case. On the other hand, "respectively and as shown in figure" is used to indicate that elements in a list correspond to elements in another list or figure. They are used in different contexts and cannot be interchanged.

Last updated: March 17, 2024 • 538 views

without loss of generality, we use

This phrase is correct and commonly used in mathematics and technical fields to simplify proofs by assuming a particular case.

This phrase is used to introduce an assumption made in a proof or argument to simplify the discussion without loss of generality.

Examples:

Without loss of generality, we assume that the function is continuous.
Without loss of generality, we consider the case where x is positive.

Show examples from the web [+]

... of light in vacuum; and V(→r) is an arbitrary external scalar potential; one could easily take it to be quadratic in x and y, without loss of generality. We use.
Without loss of generality, we use. (x1,s(x1)),…, (xk,s(xk)) as k shadows. We can reveal the secret value M by using Lagrange interpolation. where M=s(0).
Jun 3, 2015 ... We consider a variety of μ vectors: (a) the null scenario where all μi=μ0, and without loss of generality we use μ0=0; (b) a “sparse” scenario in ...
Jun 12, 2015 ... Without loss of generality, we use the time unit (min) as the generalized cost unit so that we can minimize the monitoring cost of all incidents.

respectively and as shown in figure

This phrase is correct and commonly used to indicate that elements in a list correspond to elements in another list or figure.

This phrase is used to clarify the correspondence between elements in different lists or figures, especially in technical or academic writing.

Examples:

The values 1, 2, and 3 correspond to A, B, and C, respectively, as shown in Figure 1.
The variables x, y, and z represent length, width, and height, respectively, as shown in the diagram.

Show examples from the web [+]

Three dimers are formed between molecules A and B, C and D, and E and F, respectively, and as shown in Figure 5, A and B ▷, in each case, the two ...
respectively, and as shown in Figure 2, they encompass the segments of VP16 altered by the RI-i and RII-i inser- tion mutations. To characterize the specificity of ...
Sep 9, 2008 ... respectively) and, as shown in Figure 3A, no significant increase in the rate of protrusion formation or loss could be observed over the next 24 h ...
Aug 30, 2012 ... (2005) respectively (and as shown in Figure. 1 of Kassin et al. (2007)). The vertical solid line is the same as that in Figure 2, showing the limits.

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without loss of generality, we use vs respectively and as shown in figure