Grade 7 Math Questions on Set Theory
With Step-by-Step Solutions

Welcome to the Grade 7 Set Theory study page. This platform is curated to guide students, teachers, and parents through the foundational rules of sets. Below, you will find practice questions covering set cardinality, element membership, subsets, unions, and intersections alongside comprehensive, step-by-step solutions.

Practice Questions

  1. Find the cardinality of set $A$ and set $B$ defined by:

    $A = \{a, b, c, d\}$  and  $B = \{1, 4, 7, 9, 10, 12, 23\}$

    View Solution

    The cardinality of a set is equal to the number of distinct elements contained in that set. We use the notation $|A|$ or $\text{Card}(A)$ to represent it.

    • Set $A$ has $4$ distinct elements, hence $\mathbf{|A| = 4}$
    • Set $B$ has $7$ distinct elements, hence $\mathbf{|B| = 7}$
  2. Determine the cardinality of each of the following sets:
    • The set of all months in a year beginning with the letter "M".
    • The set of all vowels in the English alphabet.
    View Solution

    a) The set of months starting with "M":
    Let this be set $S$. Listing the elements gives: $S = \{\text{March}, \text{May}\}$.
    Thus, the cardinality is $\mathbf{|S| = 2}$.

    b) The set of all vowels in the English alphabet:
    Let this be set $V$. Listing the elements gives: $V = \{a, e, i, o, u\}$.
    Thus, the cardinality is $\mathbf{|V| = 5}$.

  3. Sets $A$ and $B$ are defined by:

    $A = \{3, 5, 7, 8\}$  and  $B = \{x, y, z\}$

    Answer by True or False:
    • $3 \in A$
    • $3 \in B$
    • $x \notin A$
    • $z \in B$
    • $8 \in B$
    View Solution

    The symbol $\in$ denotes "is an element of" (membership), while $\notin$ denotes "is not an element of".

    • True; $3$ is visibly present as a member inside set $A$.
    • False; $3$ is not an element of set $B$.
    • True; the element $x$ does not belong to set $A$.
    • True; $z$ is explicitly an element of set $B$.
    • False; $8$ belongs to set $A$, but it is not an element of set $B$.
  4. List all the elements belonging to each of the following sets:
    • The set of all positive even numbers less than or equal to $10$.
    • The set of all unique letters in the word "AUSTRALIA".
    • The set of all whole numbers greater than $3$ and smaller than $16$ that are divisible by $3$.
    • The set of all whole numbers greater than $5$ and smaller than $35$ that are divisible by $5$.
    • The set of all prime numbers divisible by $3$.
    • The set of all numbers whose absolute value is equal to $7$.
    View Solution
    • $\{2, 4, 6, 8, 10\}$ (Positive even numbers starting from $2$ up to and including $10$).
    • $\{A, U, S, T, R, L, I\}$ (In set roster notation, duplicate elements like "A" are listed only once).
    • $\{6, 9, 12, 15\}$ (Multiples of $3$ lying strictly between $3$ and $16$).
    • $\{10, 15, 20, 25, 30\}$ (Multiples of $5$ lying strictly between $5$ and $35$).
    • $\{3\}$ (The only prime number divisible by $3$ is $3$ itself).
    • $\{-7, 7\}$ (Both $|-7| = 7$ and $|7| = 7$).
  5. Given the following sets:

    $A = \{2, 3, 4, 5, 6, 7\}$
    $B = \{3, 5, 7\}$
    $C = \{3, 5, 7, 20, 25, 30\}$
    $D = \{20, 25, 30\}$

    Answer by True or False:
    • $A \subseteq B$
    • $B \subseteq A$
    • $B \nsubseteq C$
    • $C \subseteq D$
    • $D \nsubseteq A$
    View Solution

    The symbol $\subseteq$ means "is a subset of" (every element of the first set must be inside the second set). The symbol $\nsubseteq$ means "is not a subset of".

    • False; elements $2, 4,$ and $6$ belong to $A$ but are missing from $B$.
    • True; every single element in set $B$ ($3, 5,$ and $7$) is also found inside set $A$.
    • False; because all elements in $B$ exist inside $C$, $B$ is a subset ($B \subseteq C$).
    • False; elements $3, 5,$ and $7$ are in set $C$ but completely missing from set $D$.
    • True; elements $20, 25,$ and $30$ are present in set $D$ but do not exist anywhere inside set $A$.
  6. Using the sets $A, B, C$, and $D$ defined in Question 5, determine the following:
    • a) $A \cup B$
    • b) $A \cap B$
    • c) $B \cap C$
    • d) $C \cup B$
    • e) $D \cap C$
    • f) $(A \cap B) \cap C$
    • g) $(A \cup B) \cap (C \cup D)$
    • h) $(A \cup B) \cup (C \cup D)$
    View Solution

    Recall that the union symbol ($\cup$) combines all elements from both sets, while the intersection symbol ($\cap$) extracts only elements common to both sets.

    • $A \cup B = \{2, 3, 4, 5, 6, 7\} = A$ (Since $B$ is completely contained within $A$).
    • $A \cap B = \{3, 5, 7\} = B$ (These are the shared items between sets $A$ and $B$).
    • $B \cap C = \{3, 5, 7\} = B$ (The shared items between sets $B$ and $C$).
    • $C \cup B = \{3, 5, 7, 20, 25, 30\} = C$ (Since $B$ is completely contained within $C$).
    • $D \cap C = \{20, 25, 30\} = D$ (Since $D \subseteq C$).
    • $(A \cap B) \cap C = B \cap C = \{3, 5, 7\} = B$ (First find $A \cap B = B$, then intersect with $C$).
    • $(A \cup B) \cap (C \cup D) = A \cap C = \{3, 5, 7\} = B$ (Evaluate parenthetical blocks: $(A \cup B) = A$ and $(C \cup D) = C$, which leaves $A \cap C$).
    • $(A \cup B) \cup (C \cup D) = A \cup C = \{2, 3, 4, 5, 6, 7, 20, 25, 30\}$ (Combines all unique elements from $A$ and $C$ together).