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Let $X$ be a set and $f$ be a function from $X$ to $\{0, 1\}$, the field with two elements. The support of $f$ is the set $f ^ {-1}$ (1), which we denote by $\DeclareMathOperator{\supp}{supp}\supp(f)$. For a collection $F$ of such functions define $\:\supp(F) = \bigcap_{f∈F} \supp(f)$

Note that when $F = ∅, \:\supp(F) = X$, and that for such functions, $f$ is the characteristic function of its support, i.e., $f = χ_{\supp(f)}$. Find the supports of $f · g$, $1 − f$, $f + g − f · g$, and $1 − f + f · g$ in terms of the supports of $f$ and $g$.

Do now know where to start. Will be helpful to get at least the first one to see how it works.

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I give you a couple of examples.

Let $f \cdot g \colon X \to \{0,1\}$ be the function defined by $(f \cdot g) (x) = f(x) \cdot g(x)$ for all $x \in X$. The support of $f \cdot g$ is the intersection of the supports of $f$ and $g$. Indeed, by definition of support, \begin{align} \operatorname{supp}(f \cdot g) &= \{x \in X \mid f(x) \cdot g(x) = 1\} \\ &= \{x \in X \mid f(x) = 1 \text{ and } g(x) = 1\} \\ &= \{x \in X \mid f(x) = 1\} \cap \{x \in X \mid g(x) = 1\} = \operatorname{supp}(f) \cap \operatorname{supp}(g). \end{align}

Let $1 \!-\! f \colon X \to \{0,1\}$ be the function defined by $(1 \!-\! f)(x) = 1 - f(x)$ for all $x \in X$. The support of $1 \!-\!f$ is the complement of the support of $f$. Indeed, by definition of support, \begin{align} \operatorname{supp}(1 \!-\! f) &= \{x \in X \mid f(x) = 0\} \\ &= \{x \in X \mid f(x) \neq 1\} \\ &= X \smallsetminus \{x \in X \mid f(x) = 1\} \ = X \smallsetminus \operatorname{supp}(f). \end{align}

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  • $\begingroup$ then will the following be true: supp(1-f + f * g) = X∖supp(f) or supp(g) ?? $\endgroup$ – Amalya Feb 16 at 14:27
  • $\begingroup$ @Amalya - What does it mean "or" in your comment? It should be a set-theoretic operation. Anyway, you had the right intuition. $\endgroup$ – Taroccoesbrocco Feb 16 at 14:29
  • $\begingroup$ I meant "union" by "or". i am getting it right, yes? $\endgroup$ – Amalya Feb 16 at 14:31
  • $\begingroup$ @Amalya - Yes, you're right. What about $\text{supp}(f + g - f \!\cdot\! g)$? $\endgroup$ – Taroccoesbrocco Feb 16 at 14:35
  • $\begingroup$ (X∖supp(f) ∩ supp(g)) ∪ (X∖supp(g) ∩ supp(f)). is it right? $\endgroup$ – Amalya Feb 16 at 15:11

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