Definition:

If A and B are two sets then, difference of set A and set B is the all elements of set A, that are not elements of set B.

or

The Difference of two Sets A and B, is set of elements, which belong to A but not to B.

Difference of Sets A and B is denoted by A – B, and read as “A minus B”.

We can rewrite the definition of difference of two sets as,

             A – B = {X : X ∈ A and X ∉ B}

A – B means elements of A which are not the elements of B.

Difference of Sets B and A is denoted by B – A, and read as “B minus A”.

We can rewrite the definition of difference of two sets as,

             B – A = {X : X ∈ B and X ∉ A

B – A means elements of B which are not the elements of A.

The difference of two subsets A and B is a subset of U, denoted by A – B and defined by

A – B = {x : x ∈ A, x ∉ B}

B – A = {x : x ∈ B, x ∉ A}

Example 1: If X = {1, 2} and Y = {1, 2, 3, 4, 5}, find the value of X – Y and Y – X

Solution: The elements in only X are {1, 2,} and elements in only Y is {3, 4, 5}.

∴ X – Y = {1, 2} and Y – X = {3, 4, 5}

Example 2: If Set A = {1, 4, 7, 8, 9} and Set B = {3, 2, 1, 7, 5}, find A – B

Solution: A – B = {4, 8, 9}

Example 3: If A = {x, y, p, q,} and B = {r, s, p, q}. Find the value of (i) A – B (ii) B – A (iii) A ∩ B

Solution: A – B = (x, y). Therefore, the elements (x, y) belongs to A but not to B, and the elements (r, s) belongs to B but not to A.

B – A = (r, s)
A ∩ B = {p, q}

Example 4: If R is a set of real numbers and P is a set of rational numbers. Find set of (R – P)

Solution: R is a set of real numbers and P is a set of rational numbers.
Therefore, set (R – P ) is a set of irrational numbers.

Example 5: If A = {a, b, c}, and B = {d, e, f}. Find the value of (i) A – B (ii) B – A

Solution: A – B = {a, b, c} = A

B – A = {d, e, f} = B

∴ A – B = A, and B – A = B

If A and B are two disjoint sets as they do not have any common element, then,

A – B = A, and B – A = B

Example 6: If three sets A = {10, 11, 12, 13, 14}, B = {8, 10, 12, 14, 16} and C {5, 7, 9, 11, 14, 16}. Find the difference of sets and represents the sets in Venn diagram.
(i) A – B
(ii) B – A
(iii) C – A
(iv) B – C

Solution: Given that A = {10, 11, 12, 13, 14}, B = {8, 10, 12, 14, 16} and C = {5, 7, 9, 11, 14, 16}

∴ A – B = {11, 13}

∴ B – A = {8, 16}

∴ C – A = {5, 7, 9, 16}

∴ B – C = {8, 10, 12}

Identities: Difference of sets

(i) If set A and B are equal then,

A – B = A – A = ∅ (empty set)

(ii) If we subtract an empty set is from a set (suppose set is A) then, the result is that set itself, i.e.

A – ∅ = A

(iii) If we subtract a set from an empty set then, the result is an empty, i.e.

∅ – A = ∅

(iv)

A – U = ∅