Two sets can overlap, sit side by side, or one can swallow the other. Intersection and union name the two ways of combining them, and a Venn diagram shows which case you are in.
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
Of 10 students, 6 study music and 5 study art. Can you tell how many study both? If not, what extra information would you need?
Before you work it out, what is your instinct? Write it down, then check it against the lesson.
Work through the core explanation before applying it.
The intersection $A \cap B$ is the elements in $A$ and in $B$. The union $A \cup B$ is the elements in $A$ or in $B$, including those in both. If nothing is shared, $A \cap B = \varnothing$ and the sets are disjoint.
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
Work through the core explanation before applying it.
The intersection $A \cap B$ collects the elements appearing in both sets. For $A = \{1,2,3\}$ and $B = \{2,3,4\}$: $A \cap B = \{2,3\}$.
The union $A \cup B$ collects everything in either set. For the same two: $A \cup B = \{1,2,3,4\}$. Notice 2 and 3 appear once each, because a set never repeats an element.
In everyday speech or often means one but not the other. In mathematics it always includes both, so $A \cup B$ contains the overlap as well.
$A \cap B$ is the elements in both sets, $A \cup B$ the elements in at least one. Mathematical or is inclusive, so the union contains the overlap. A set never lists an element twice.
Pause, copy both definitions with the worked example, and the note that $\cap$ reads and while $\cup$ reads or, into your book.
We just saw what happens when two sets overlap. That raises a question: what if they share nothing at all? This card answers it → they are called disjoint, and their intersection is the empty set.
Two sets are disjoint when they have no elements in common, that is when $A \cap B = \varnothing$. The odd numbers and the even numbers are disjoint.
On a Venn diagram, disjoint sets are drawn as two circles that do not touch, because there is no overlap region to draw.
For disjoint sets only, $n(A \cup B) = n(A) + n(B)$, because nothing has been counted twice. That is a special case of the counting rule in the next lesson.
Disjoint sets share no elements, so $A \cap B = \varnothing$ and their circles do not overlap. For disjoint sets alone, $n(A \cup B) = n(A) + n(B)$. In probability the same idea is called mutually exclusive.
Pause, copy the definition of disjoint, the picture of two separate circles, and the special case, into your book.
We just saw the three ways two sets can sit relative to each other. That raises a question: how do you get worded information into a diagram without double counting? This card answers it → start in the middle and work outwards.
Draw the universal set as a rectangle and each set as a circle inside it. Two overlapping circles create four regions: $A$ only, both, $B$ only, and neither.
Fill the overlap first. If 30 students study music, 20 study art and 8 study both, put 8 in the middle. Then music only is $30 - 8 = 22$ and art only is $20 - 8 = 12$.
If the universal set has 50 students, the outside region is $50 - (22 + 8 + 12) = 8$ who study neither. Every student is now in exactly one region, which is the check that the diagram is right.
Draw the rectangle and circles, fill the overlap first, subtract it from each circle total, then find the outside region by subtracting from $n(\xi)$. Every element ends in exactly one region, and the regions must total $n(\xi)$.
Pause, copy the four regions, the fill-the-middle-first order with the music and art example, and the totals check, into your book.
Worked examples · 3 in a row, reveal as you go
Follow the reasoning through complete worked solutions.
Let $A = \{1,2,3,4\}$ and $B = \{3,4,5\}$. Find $A \cap B$, $A \cup B$ and $n(A \cup B)$.
In a group of 50 students, 30 study music, 20 study art and 8 study both. How many study neither?
Let $P$ be the prime numbers under 10 and $E$ the even numbers under 10 excluding 2. Are $P$ and $E$ disjoint?
Work through the core explanation before applying it.
Run the quick drill and copy the summary into your book.
At the start you were told 6 of 10 students study music and 5 study art, and asked whether you could find how many study both. Explain why you could not, and say what the smallest possible overlap was and why.
Answer the drill bank and rate your confidence.
Pick your answer, then rate your confidence, that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
Write full responses, then check them against the model answers.
Q1. In a survey of 80 people, 45 own a dog, 32 own a cat and 15 own both. Draw a Venn diagram showing all four regions, and state how many own neither. (4 marks)
Q2. Let $A = \{2,4,6,8\}$ and $B = \{1,2,3,4\}$. Find $A \cap B$, $A \cup B$ and $n(A \cup B)$. (3 marks)
Q3. Explain why $n(A \cup B)$ is usually less than $n(A) + n(B)$, and state the one case where they are equal. (2 marks)
Practice 1: $\{2,5\}$. Practice 2: $\{1,2,3\}$. Practice 3: yes, no shared elements. Practice 4: $40 - 33 = 7$.
Q1 (4 marks): Middle region 15 [1]. Dog only $45 - 15 = 30$; cat only $32 - 15 = 17$ [1]. Total inside the circles $= 62$ [1]. Neither $= 80 - 62 = 18$ [1].
Q2 (3 marks): $A \cap B = \{2, 4\}$ [1]. $A \cup B = \{1,2,3,4,6,8\}$ [1]. $n(A \cup B) = 6$ [1].
Q3 (2 marks): Adding $n(A)$ and $n(B)$ counts every shared element twice, once in each set, so the sum overstates the union by the size of the overlap [1]. They are equal only when the sets are disjoint, since then there is no overlap to double count [1].
Take the module quiz if you are ready, then mark the lesson complete.
Find intersections and unions and fill Venn diagrams from worded information. Beat the boss to bank a tier, gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.
⚔ Enter the arenaTick when you've finished the practice and review.
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