Sets — Practice

Question 1

Write the set A = {x : x is an integer, x^2 = 9} in roster form, and B = {5, 10, 15, 20, 25} in set-builder form.

Accepted Answer

A: x^2=9 gives x=3 or x=-3, so A={-3,3}. B: each element is a multiple of 5 up to 25, so B={x : x=5n, n in N, n<=5}.

Question 2

Classify each as empty, finite, or infinite: (i) {x : x is a natural number and x+3=2}, (ii) the set of lines through a fixed point in a plane, (iii) {x : x is an even prime number}.

Accepted Answer

(i) x+3=2 gives x=-1, which is not a natural number, so this set is empty. (ii) infinitely many lines pass through any single point, so this is infinite. (iii) the only even prime is 2, so this set is finite, with exactly one element, {2}.

Question 3

How many subsets and how many proper subsets does a set with 5 elements have? List one subset containing exactly 2 elements as an example.

Accepted Answer

Total subsets = 2^5 = 32. Proper subsets exclude the set itself, so 32-1 = 31. For a set {a,b,c,d,e}, an example 2-element subset is {a,c}.

Question 4

Write {x : x is a real number, -2 <= x < 5} as an interval, and write the interval (0, 10] in set-builder form.

Accepted Answer

{x : -2<=x<5} = [-2, 5), since the left endpoint is included and the right is not. (0,10] = {x in R : 0<x<=10}, since the round bracket at 0 excludes it and the square bracket at 10 includes it.

Question 5

If A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7}, find A union B, A intersection B, and state whether A and B are disjoint.

Accepted Answer

A union B = {1,2,3,4,5,6,7}. A intersection B = {4,5}. Since the intersection is not empty, A and B are NOT disjoint.

Question 6

If A = {2, 4, 6, 8, 10} and B = {4, 8, 12}, find A - B and B - A, and explain why they are different sets.

Accepted Answer

A-B = {2,6,10} (A's elements not in B). B-A = {12} (B's elements not in A). They differ because A-B keeps only A's own leftover elements while B-A keeps only B's -- the two operations filter from opposite starting sets.

Question 7

If U = {1,2,...,10}, A = {1,2,3,4,5}, B = {4,5,6,7}, find A' and B', and hence find A' union B'.

Accepted Answer

A' = U-A = {6,7,8,9,10}. B' = U-B = {1,2,3,8,9,10}. A' union B' = {1,2,3,6,7,8,9,10} (every element that fails to be in A or fails to be in B).

Question 8

If U = {1,2,...,9}, A = {1,3,5,7}, B = {2,3,5,8}, verify De Morgan's second law: (A intersection B)' = A' union B'.

Accepted Answer

A intersection B = {3,5}, so (A intersection B)' = {1,2,4,6,7,8,9}. Separately, A' = {2,4,6,8,9}, B' = {1,4,6,7,9}, so A' union B' = {1,2,4,6,7,8,9}. Both sides match, verifying the law.

Question 9

Let A = {2, {3}, 4, {5, 6}}. State, with reasons, whether each is true: (i) 3 in A, (ii) {3} in A, (iii) {3} subset A, (iv) {5,6} subset A.

Accepted Answer

(i) False: 3 itself is not one of A's four listed elements (only the set {3} is). (ii) True: {3} is literally one of A's elements. (iii) True: since {3} is an element of A, the set containing just it, {{3}}, is a subset -- but written as {3} subset A this reads as asking whether the set {3} itself is a subset, which requires 3 to be an element of A, which is false, so actually this is FALSE; the distinction between (ii) and (iii) is exactly the trap. (iv) False: 5 and 6 individually are not elements of A, only the set {5,6} is.

Practice — Sets

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