Question Video: Finding the Union of Two Sets of Numbers Involving Interval Notation | Nagwa Question Video: Finding the Union of Two Sets of Numbers Involving Interval Notation | Nagwa

Question Video: Finding the Union of Two Sets of Numbers Involving Interval Notation Mathematics • Second Year of Preparatory School

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Given that 𝑋 = [−7, −6] and 𝑌 = [3, ∞), find 𝑋 ∪ 𝑌.

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Video Transcript

Given that 𝑋 is equal to the closed interval from negative seven to negative six and 𝑌 is equal to the left-closed right-open interval from three to ∞, find 𝑋 union 𝑌.

In this question, we are given two sets, 𝑋 and 𝑌, in interval notation. And we need to find the union of these two sets. To do this, we can start by recalling that a union of sets means that we take all of the elements in both sets. So, we say that 𝑎 is a member of the union of 𝑋 and 𝑌 if 𝑎 is a member of 𝑋 or if 𝑎 is a member of 𝑌.

To find the union of these two intervals, we should start by determining exactly which numbers are members of each set. First, we note that 𝑋 is a closed interval. So, the endpoints of the interval are included. This means we want to include all real numbers between negative seven and negative six including the endpoints. We can follow a similar process for 𝑌. We note that 𝑌 is closed at three and unbounded above. So, the set 𝑌 includes all real values greater than or equal to three.

At this point, we could find the union of 𝑋 and 𝑌 using set builder notation. However, when working with intervals, it is often easier to visualize the sets using a number line first. First, 𝑋 is the set of real numbers between negative seven and negative six, including the endpoints. We can represent this on a number line using solid dots at negative seven and negative six to show that they are included in the set.

We can apply a similar process to 𝑌. We want to include all of the values greater than or equal to three in the set. So, we sketch a solid dot at three and a line to the right of three. We include an arrow on the right to show that the set is not bounded above.

There are a few different ways of finding an expression for the union of these sets. One way is to find a set that includes both 𝑋 and 𝑌. For instance, we can see that the set of all real values greater than or equal to negative seven encompasses both 𝑋 and 𝑌.

However, we can see that this set includes all of the real values between negative six and three, which are not elements of 𝑋 or 𝑌. So, we need to remove these values from our set to find the union of 𝑋 and 𝑌. This means that we want to remove the open interval from negative six to three from our set to find the union of 𝑋 and 𝑌.

Finally, we can recall that we can remove the elements of a set from another set by using the set minus operation. This means that we can write the union of 𝑋 and 𝑌 as the left-closed, right-open interval from negative seven to ∞ minus the open interval from negative six to three. It is also worth noting that we can use parentheses or reversed brackets to show that an interval does not include the endpoints. Both notations are common, and it is personal preference which one to use.

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