Solved Problems on Difference of Sets

Problem 1. Find the difference W – N, where W is the set of whole numbers and N is the set of natural numbers.

Solution:

Step 1. Write the given sets in set- builder form.

W = {0, 1, 2, 3, 4, 5,….,∞}

N = {1, 2, 3, 4, 5,….,∞}

W – N implies all elements of set W and none of set N.

Step 2. Write the difference in mathematical form: W – N = {0, 1, 2, 3, 4, 5,….,∞} – {1, 2, 3, 4, 5,….,∞}

Step 3. Strike out all the common elements in both W and N. {0, 1, 2, 3, 4, 5,….,∞} – {1, 2, 3, 4, 5,….,∞}.

All elements left in W represent the difference W – N.

Hence, W – N = {0}.

Problem 2. Prove  P – (Q ∪ R) = (P – Q) ∩ (P – R), if P = {1, 2, 4, 5}; Q = {2, 3, 5, 6} and R = {4, 5, 6, 7}.

Solution:

Let us consider the LHS first.

(Q ∪ R) = {x: x ∈ Q or x ∈ R}

⇒ Q ∪ R = {2, 3, 4, 5, 6, 7}.

Since P – (Q ∪ R) can be expressed as {x ∈ P: x ∉ (Q ∪ R)}.

⇒ P – (Q ∪ R) = {1}

Let us consider the RHS now.

P – Q is defined as {x ∈ P: x ∉ Q}

P = {1, 2, 4, 5}

Q = {2, 3, 5, 6}

⇒ P – Q = {1, 4}

Now, P – R is defined as {x ∈ P: x ∉ R}

⇒ P – R = {1, 2}

(P – Q) ∩ (P – R) = {x: x ∈ (P – Q) and x ∈ (P – R)}.

= {1}

∴ LHS = RHS

Hence verified.

Problem 3. If S and T are two sets, prove that: (S ∪ T) – T = S – T.

Solution:

Let us consider LHS first.

(S ∪ T) – T

= (S – T) ∪ (T – T)

= (S – T) ∪ ϕ (since, T – T = ϕ)

= S – T (since, x ∪ ϕ = x for any set)

= RHS

Hence proved.

Problem 4. If n(S) = 69, n(T) = 55, and n(S ∩ T) = 10, then what is n(S Δ T)?

Solution:

Since, n(S U T) = n(S) + n(T) – n(S∩ T)

= 69 + 55 – 10

= 114

According to the symmetric difference of sets,

n(S Δ T) = n(S U T) – n(S ∩ T)

= 114 – 10

n(S Δ T) = 104

Problem 5. If P, Q, R are three sets, such that P ⊂ Q, then prove that R – Q ⊂ R – P.

Solution:

Given, P ⊂ Q

To prove: R – Q ⊂ R – P

Let us consider, x ∈ R – Q

⇒ x ∈ R and x ∉ Q

⇒ x ∈ R and x ∉ P

⇒ x ∈ R – P

Thus, x ∈ R – Q ⇒ x ∈ R – P

This is true for all x ∈ R – Q

∴ R – Q ⊂ R – P

Hence proved.

Difference of Sets is the operation defined on sets, just like we can perform arithmetic operations on numbers in mathematics. Other than difference we can also perform union and intersection of sets for any given sets. These operations have a lot of important applications in mathematical practice. In this article, we will learn about the difference of sets including its definition, mathematical expressions, Venn diagram as well as properties of difference of sets. Let’s start our learning of the “Difference of Sets”.