Contradiction of "For all $𝑎, 𝑏, 𝑐, 𝑑 ∈ ℤ,$ if $𝑎\mid𝑏$ and $(𝑎𝑐) ∤ (𝑏𝑑)$, then $𝑐 ∤ 𝑑.$"
39953
$a\mid b,\ c\mid d\,\Rightarrow\ ac\mid bd $ $\ \, \bf\small [Divisibility\ Product\ Rule]$
75387
A proof that $1=2$. May I know why it’s false?
49453
Why do I have to solve for $x$ in this proof, if I choose not to find a counterexample?
52614
(exercise from Tao's analysis book) Proof of a lemma relating to power set of X
20658
Difficulty in finding a counterexample
59724
Proving that $x_n\to L$ implies $|x_n|\to |L|$, and what about the converse?
106342
Why $L$-structures cannot satisfy a contradiction
41318
Can every proof by contradiction also be shown without contradiction?
132701
Contradicting Fubini's theorem
130455
If $H,K⊲G$ and $H∩K = \{1_G\}$, then all elements in $H$ commute with all elements in $K$
28007
Assuming: $\forall x \in [0,1]:f(x) > x$ Prove: $\forall x \in [0,1]:f(x) > x + \varepsilon $
32525
If $f(1)=1$, then is it true that $f(n)=n$ for all $n \in \mathbb{N}\cup\{0\}$.
5293
Show that the logic $ \ \forall x \exists y \ P(x,y) \ $ is not logically equivalent to $ \ \exists x \forall y \ P(x,y) \ $
142026
Show that $A∩B∩C= ∅$ is only true when $A∩B = ∅, A∩C = ∅$ or $B∩C = ∅$ or show a counterexample.
111756
How do I prove that then $f(x) = 1$ for some $x ∈ [0, 1]$?
6166
Is the following statement provable?
20789
If $\Sigma \vdash \varphi$ then $\Sigma \vdash \forall x \varphi$. Why when $(\varphi \implies \forall x \varphi)$ is not true?