Indirect proof problems, solved and explained

Recent questions in Indirect Proof

Cheyanne Jefferson 2022-08-16

Can we prove $\neg R (x) ⟹ \neg R (y)$ by proving $\neg \neg R (y) ⟹ \neg \neg R (x)$ ?
Can we also prove that by proving $R (y) ⟹ R (x)$ ?

assuolareuz 2022-08-14

Give a direct proof that the set ${x | Φ_{x} (1) ↓}$ (which is a set of program numbers that halt on input $1$ ) is not recursive.

Ebone6v 2022-08-13

Suppose $R$ is a partial order on a set $A$ , and $B \subseteq A$ .
Prove that, if $R$ is a total order and $b$ is a minimal element of $B$ , then $b$ is the smallest element of $B$ .

pigskiniv 2022-08-12

In an indirect proof, is it possible to reject the assumption based on contradiction of a premise?

yongenelowk 2022-08-11

$\begin{array}{l} Theorem 2 - 17 : Let {a_{n}} be a bounded sequence of real numbers. Then: \\ \begin{array}{lll} (a) lim sup a_{n} = L if and only if, for any ε > 0, there are infinitely many terms of \\ {a_{n}} in (L - ε, L + ε) but only finitely many terms of {a_{n}} with a_{n} > L + ε . \end{array} \end{array}$
Proves directly that $lim sup {a_{n} + b_{n}} \leq lim sup {a_{n}} + lim sup {b_{n}}$

Matonya 2022-08-11

Show that
$T (X) = (R, V) = (X_{(n)} - X_{(1)}, \frac{X_{(n)} + X_{(1)}}{2})$
is a minimal sufficient statistic for $θ$ .

Rose Graves 2022-08-10

Is it possible to derive Peirce's Law:
$⊢ * [(α \to β) \to α] \to α$
in a calculus that has modus ponens, the Deduction Theorem, Cut rule, Inconsistency effect and Principle of Indirect Proof?

Leia Hood 2022-08-09

Proof
$5 ∣ n ⟺ 5 ∣ n^{2}$

Jaylyn Gibson 2022-08-06

$A ∖ B = B ∖ A ⟺ A = B$
can we actually prove it without using contradiction?

Stephanie Hunter 2022-07-19

$\forall k, l \in Z, k l is even ⟹ k is even \lor l is even .$
Suppose $k$ is odd and $l$ is odd, then there exist integers $a$ and $b$ such that $k = 2 a + 1$ , so $k l = (a b ? ?)$

Jadon Melendez 2022-07-17

How to solve this type of sum using indirect proof. For any real number $x$ , if $x^{3} + 2 x + 33 \neq 0$ , then $x + 3 \neq 0$

kadejoset 2022-07-14

Theorems that we can prove only by contradiction