without loss of generality
(mathematics) With a constraining assumption that, however, makes it clear how to apply the proof performed under this assumption to the general case unconstrained by the assumption.
We wish to prove two ellipses in the plane intersect in an even number of points or are tangent. Without loss of generality, assume one of them has the equation
x
2
+
y
2
=
1
{\displaystyle x^{2}+y^{2}=1}
.
Proof Without loss of generality, consider the parabola Suppose that two tangents contact this parabola at the points and Their slopes are and respectively. Source: Internet
Without loss of generality, assume it uses only AND and NOT gates. Source: Internet
Without loss of generality, assume that f(x) is normalized: : It follows from the Plancherel theorem that is also normalized. Source: Internet
Without loss of generality, it may be specified that v is normalized so that the sum of its four components is unity. Source: Internet
Without loss of generality, take p 1 < q 1 (if this is not already the case, switch the p and q designations.) Consider : and note that 1 < q 2 ≤ t < s. Therefore t must have a unique prime factorization. Source: Internet
Without loss of generality, we may assume that h(x) ≥ 0 (otherwise, choose −x instead). Source: Internet