Question

Let A = { x ∈ R : x 2 − 5 x + 4 ≤ 0 } , B = (3 , 5), and C = (3 , 4]. Show that A ∩ B = C . (Recall, you must show two separate statements: A ∩ B ⊆ C and C ⊆ A ∩ B . Hint, you may benefit from factoring the polynomials that appear in the sets.)

Answer 1

You can proceed as follows:

Explanation:

First solve the quadratic inequality
`x^(2)-5x+4\leq 0`. To do that, factorize, then we have that
`(x-4)(x-1)\leq 0`. This implies that

`x-1\leq 0\, \text{and}\, x-4\geq 0`

or

`x-1 \geq 0\, \text{and}\, x-4\leq 0`

In the first case the solution is the empty set
`\emptyset`. In the second case the solution is the interval
`1\leq x \leq 4`. Now we have that

`A=[1,4]`

`B=(3,5)`

`C=(3,4]`.

To show that
`A\cap B\subseteq C` consider
`x\in A\cap B`. Then
`1\leq x \leq 4\, \text{and}\, 1<x<5`, this implies that
`3<x\leq 4`, then
`x\in C`. Now, to show that
`C\subseteq A\cap B` consider
`x\in C`, then
`3<x\leq 4`, then
`1\leq x \leq 4\, \text{and}\, 3<x<5`, then
`x\in [1,4] \, \text{and}\, x\in (3,5)`, this implies that
`x\in A\cap B`.

Observe the image below.