Q.A die is thrown. Describe the following events:

Concept: Set Operations. With one die, $S=\{1,2,3,4,5,6\}$, and every event is a subset of $S$.

Writing each event as a set:

• $A=\{1,2,3,4,5,6\}=S$ (all faces are $<7$)
• $B=\varnothing$ (no face is $>7$)
• $C=\{3,6\}$, $\;D=\{1,2,3\}$, $\;E=\{6\}$ (even and $>4$), $\;F=\{3,4,5,6\}$ (not less than $3$)

Applying the operations (note $F'=\{1,2\}$):

• $A\cup B=\{1,2,3,4,5,6\}$, $\;A\cap B=\varnothing$, $\;B\cup C=\{3,6\}$
• $E\cap F=\{6\}$, $\;D\cap E=\varnothing$
• $A-C=\{1,2,4,5\}$, $\;D-E=\{1,2,3\}$
• $E\cap F'=\{6\}\cap\{1,2\}=\varnothing$, $\;F'=\{1,2\}$

When a die is thrown once, the possible outcomes form the sample space

Every event below is a subset of $S$, so we only ever use the numbers $1$ to $6$ — the die can never show a $0$ or a $7$.

Step 1 — Describe each event as a set

• A: a number less than 7. Every face is less than $7$, so $A=\{1,2,3,4,5,6\}=S$.
• B: a number greater than 7. No face exceeds $7$, so $B=\varnothing$ (the empty set).
• C: a multiple of 3. The multiples of $3$ up to $6$ are $3$ and $6$, so $C=\{3,6\}$.
• D: a number less than 4. These are $1,2,3$, so $D=\{1,2,3\}$.
• E: an even number greater than 4. The even faces are $2,4,6$; only $6$ is greater than $4$, so $E=\{6\}$.
• F: a number not less than 3. "Not less than $3$" means $3$ or more, so $F=\{3,4,5,6\}$.

Step 2 — Carry out each operation

• $A\cup B$: union with the empty set leaves $A$ unchanged, so $A\cup B=\{1,2,3,4,5,6\}$.
• $A\cap B$: intersection with the empty set is empty, so $A\cap B=\varnothing$.
• $B\cup C$: union of the empty set with $C$ is just $C$, so $B\cup C=\{3,6\}$.
• $E\cap F$: $E=\{6\}$ and $F=\{3,4,5,6\}$ share only $6$, so $E\cap F=\{6\}$.
• $D\cap E$: $D=\{1,2,3\}$ and $E=\{6\}$ have nothing in common, so $D\cap E=\varnothing$.
• $A-C$: remove $3$ and $6$ from $A$, giving $A-C=\{1,2,4,5\}$.
• $D-E$: $6\notin D$, so nothing is removed; $D-E=\{1,2,3\}$.
• $F'$: the complement of $F=\{3,4,5,6\}$ in $S$ is $F'=\{1,2\}$.
• $E\cap F'$: $E=\{6\}$ and $F'=\{1,2\}$ share nothing, so $E\cap F'=\varnothing$.