   Chapter 1.4, Problem 6TFE

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# True or FalseLabel each of the following statements as either true or false.Let A = { a , b , c } . The empty set ∅ is the identity element in power set ℘ ( A ) with respect to the binary operation ∪ .

To determine

Whether the statement, “The empty set is the identity element in power set (A) with respect to the binary operation ” is true or false where A={a,b,c}.

Explanation

Formula Used:

An element e in set A is called an identity element with respect to the binary operation if e has the property that

ex=xe=x

The power set is defined as the set which contains all the subsets of a set.

A null set is set which contains no elements in it denoted as or{}.

Explanation:

Power set of A is (A)={,{a},{b},{c},{a,b},{b,c},{a.c},{a,b,c}} that contains all subsets of A including A.

Consider an element from power set (A) as {b}. Then,

{b}={b}={b}={b}

Here, {b}(A).

Consider another element from power set (A) as {a}. Then,

{a}={a}={a}={a}

Here, {a}(A).

Consider another element from power set (A) as {c}. Then,

{c}={c}={c}={c}

Here, {c}(A).

Consider another element from power set (A) as . Then,

==

Here, (A)

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