Relations and Functions

Question

State whether each of the following statement is true or false. Justify your answer.

(i) {2, 3, 4, 5 } and {3, 6} are disjoint sets. (ii) {a, e, i, o, u} and {a, b, c, d }are disjoint sets. (iii) {2, 6, 10, 14} and {3, 7, 11, 15} are disjoint sets. (iv) {2, 6, 10} and {3, 7, 11} are disjoint sets

Solution

(i) A = {2, 3, 4, 5}, B = {3, 6}
$space space space space rightwards double arrow space space straight A space intersection space straight B space equals space left curly bracket 2 comma space 3 comma space 4 comma space 5 right curly bracket space intersection space left curly bracket 3 comma space 6 right curly bracket space equals space left curly bracket 3 right curly bracket space not equal to straight ϕ$

∴ A and B are not disjoint sets.
(ii) A = {a, e, i, o, u}, B = {a, b, c, d}
$space space space space space space rightwards double arrow space straight A space intersection space straight B space equals space left curly bracket straight a comma space straight e comma space straight i comma space straight o comma space straight u right curly bracket space intersection space left parenthesis straight a comma space straight b comma space straight c comma space straight d right curly bracket space equals space left curly bracket straight a right curly bracket space equals straight ϕ$

∴ A and B are not disjoint sets.
(iii) A = {2, 6, 10, 14}, B = {3, 7, 11, 15}
$space space space space space rightwards double arrow space straight A space intersection space straight B space equals space left curly bracket 2 comma space 6 comma space 10 comma space 14 right curly bracket space intersection space left curly bracket 3 comma space 7 comma space 11 comma space 15 right curly bracket space equals space left curly bracket space right curly bracket space equals space straight ϕ$
Hence, A and B are not disjoint sets.
(iv) A = {2, 6, 10}, B = {3, 7, 11}
$space space space space space space space rightwards double arrow space space straight A space intersection space straight B space equals space left curly bracket 2 comma space 6 comma space 10 right curly bracket space intersection space left curly bracket 3 comma space 7 comma space 11 right curly bracket$
Hence, A and B are not disjoint sets.

Numbers	Closed under
	Addition	Subtraction	Multiplication	Division
Rational numbers	Yes	Yes		No
Integers		Yes		No
Whole numbers	...		Yes
Natural numbers		No

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