Sentence examples for purpose of contradiction from inspiring English sources

Next, assume that f(P) ⊗ I ≽ B ⊗ C, but for the purpose of contradiction that f(P) ≽ ̸λmax(C B.

To this end, suppose for the purpose of contradiction that there exists another fixed point z ∈ X of f with z ≠ x ∗.

To see this, take x = 1 and y = 0. Assume for the purpose of contradiction that there exists α ¯ : [ 0, 1 ] × [ 0, 1 ] → [ 0, 1 [ which holds all the requirements in Definition 8.

Indeed, for the purpose of contradiction, assume that there exists a sequence ( f k ) k ∈ N in C c which converges to f ∈ C c with respect to T ( p C c ) and, in addition, the sequence ( Ψ T ( f k ) ) k ∈ N does not converge to Ψ T ( f ) with respect to T ( p C c ).

Indeed, assume for the purpose of contradiction that there exists α ¯ : [ 0, 1 ] × [ 0, 1 ] → [ 0, 1 [ satisfying the requirements in Definition 7 such that | x 2 2 − y 2 2 | ≤ α ¯ ( x, y ) | x − y |. for all x, y ∈ [ 0, 1 ] (note that p max s ( x, y ) = | x − y | for all x, y ∈ [ 0, 1 ] ).

For the purpose of contradiction, assume that p > 0. Then for all n ∈ N, we have 0 < p ≤ p ( x n, x n + 1 ) ≤ p ( x n − 1, x n ) ≤ ⋯ ≤ p ( x 0, x 1 ), and from the definition of θ = θ ( p, p ( x 0, x 1 ) ), we obtain that α ¯ ( x n − 1, x n ) ≤ θ.

For the purpose of removing the contradiction between rapidity and accuracy, a variable gradient order scheme is designed for the FOLMS algorithm.

For the purpose of deriving a contradiction, we suppose that there exists a subsequence ({k_{i}}) of successful or acceptable iterations such that begin{aligned} biglVert nabla f_{k_{i}}^{T}h_{f_{k_{i}}} bigrVert geqslantepsilon^{2}_{1} >0 end{aligned} (30) for some (epsilon_{1}>0) and for all i.

As a matter of principle, though, he declines to chop stories into pieces, reassigning parts to "J" or parts to "P" for the purpose of resolving apparent contradictions.

The hope that safety might be found, as in a therapist's office, in a classroom where literature is being taught is in direct contradiction to one purpose of literature, which is to give expression through art to difficult and discomfiting ideas, and thereby to enlarge the reader's experience and comprehension.

A system which is in complete contradiction to its original purpose of being.