L4 · Sets on the number line; unions and intersections (foundations)

Learning goal

“I can compute the union and intersection of two sets, graph a set of integers or reals on a number line, and describe a graphed set in set-builder notation.”

Union — A ∪ B is the set of elements that are in A, in B, or in both. Each element is listed ONCE, even if it appears in both sets.

Intersection — A ∩ B is the set of elements that are in BOTH A and B.

Disjoint sets — two sets whose intersection is empty: A ∩ B = ∅. They share nothing.

Number-line graph of a set — a picture of a set: separate dots for a set of integers; a solid segment for a set of real numbers (every point in between is included).

Worked example

Let A = {1, 2, 3, 4} and B = {3, 4, 5, 6}. Step 1: The intersection is the overlap: elements in BOTH: A ∩ B = {3, 4}. Step 2: The union is everything from either set, each listed once: A ∪ B = {1, 2, 3, 4, 5, 6}. Note 3 and 4 appear in both sets but are written ONCE in the union. Step 3: Check the sizes: |A ∪ B| = |A| + |B| - |A ∩ B| = 4 + 4 - 2 = 6. Say it plainly: ∪ reads "or" (in A or B or both); ∩ reads "and" (in A and B).

Check yourself — explain it

Why is 3 written only once in A ∪ B?

Paired problem — example still visible

C = {2, 4, 6}, D = {4, 5, 6}: find C ∪ D and C ∩ D -> C ∪ D = {2, 4, 5, 6}; C ∩ D = {4, 6}.

Faded completion — last step is yours

"For A = {1, 2, 3} and B = {3, 4, 5}: A ∩ B = {___} and A ∪ B = {___, ___, ___, ___, ___}." -> {3}; {1, 2, 3, 4, 5}

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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