# Discrete math functions (help)

Let $$A=P(\{1,2,3,4\})$$. Let $$h$$ be the following function.

$$h : \mathbb N \to A$$ defined by $$h(x) = \{2,3\}\cap \{x\}.$$

Write down $$h(1)$$

I'm a bit puzzled by this question. Does this just mean that $$x = 1$$, therefore, there is no intersection or is there an actual answer to this question?

• It’s the empty set Apr 16, 2019 at 9:27
• Yes, $h(1)$ means that you substitute $x=1$. But there still is an actual answer to the question, and that is $\{\}$.
– 5xum
Apr 16, 2019 at 9:36

$$X\cap Y$$ means "The set of all elements which are both in $$X$$ and in $$Y$$". In the case of $$h(1)$$, we have $$h(1) = \{2, 3\}\cap \{1\}$$ There are no elements which are in both $$\{2, 3\}$$ and in $$\{1\}$$, so the result of that intersection is $$\varnothing = \{\phantom a\}$$.

And yes, in this case (as in most concrete cases), $$h(1)$$ does mean "Take any place in the definition of $$h(x)$$ where $$x$$ appears, exchange it with a $$1$$, and then calculate."

Hint

$$A = \mathcal P( \{ 1,2,3,4 \})$$ is the power-set of set $$A$$,

i.e. the set of subsets of $$A$$.

$$h : \mathbb N → A$$, defined by $$h(x) = \{ 2,3 \} \cap \{ x \}$$,

is a function that assign to each natural number $$n$$ a set: the intersection of the set $$\{ 2,3 \}$$ with the singleton set $$\{ x \}$$.

Thus, we may perform some simple checks : with $$x=2$$ we have that $$h(2) = \{ 2,3 \} \cap \{ 2 \} = \{ 2 \}$$.

Now, what about $$x=1$$ ?