query: If $f$ is an injection, $f(S_1 \cap S_2) = f(S_1) \cap f(S_2)$
positive_passages.docid: 25997
positive_passages.text: Conditions Equivalent to Injectivity
negative_passages.docid: 51395
negative_passages.text: $f\in L^1(\mathbb{R}) \cap C^1(\mathbb{R})$ and $f'\in L^1(\mathbb{R})$
negative_passages.docid: 152841
negative_passages.text: I want to show that $f(x)=x.f(1)$ where $f:R\to R$ is additive.
negative_passages.docid: 80876
negative_passages.text: Proof of $A\cap(B\cup C) = (A\cap B)\cup(A\cap C)$
negative_passages.docid: 154001
negative_passages.text: $(g\circ f)^{-1}=f^{-1}\circ g^{-1}$
negative_passages.docid: 155806
negative_passages.text: (Olympiad Question) $f(2x) + 2f(y) = f(f(x+y))$
negative_passages.docid: 40604
negative_passages.text: Is this proof correct for : Does $F(A)\cap F(B)\subseteq F(A\cap B) $ for all functions $F$?
negative_passages.docid: 91947
negative_passages.text: Proof of $ f:I_n \rightarrow I_m \Rightarrow n = m$
negative_passages.docid: 48291
negative_passages.text: Proof that $\mathrm{ker}(f^*) = \mathrm{im}(f)^{\perp} $
negative_passages.docid: 33524
negative_passages.text: Prove the following $A \setminus (B\setminus C) = (A\setminus B)\cap (A \cup C)$
negative_passages.docid: 49143
negative_passages.text: Showing $A \subset B \iff A\cap B=A$
negative_passages.docid: 75009
negative_passages.text: $A = \bigcap_{\mathfrak{p} \in \text{Spec(A)}} A_{\mathfrak{p}} = \bigcap_{\mathfrak{m} \in \text{MaxSpec(A)}} A_{\mathfrak{m}}$
negative_passages.docid: 114575
negative_passages.text: If $\sum_{i=1}^k \dim F_i > n(k-1)$ then $\cap_{i=1}^k F_i \neq \{0\}$
negative_passages.docid: 145600
negative_passages.text: Is it true that $ U \oplus W_1 = U \oplus W_2 \implies W_1 = W_2 $?
negative_passages.docid: 40586
negative_passages.text: There exists an injection from $X$ to $Y$ if and only if there exists a surjection from $Y$ to $X$.
negative_passages.docid: 152438
negative_passages.text: Is the composition of two injective functions injective?
negative_passages.docid: 28798
negative_passages.text: Proving injection and surjection with functions $F:C^B\to C^A$, $F(f)=f\circ\varphi$, $\varphi: A\to B$