Question

7. Let A = The set of all natural numbers less than 8, B = The set of all prime numben
less than 8, C = The set of even prime number. Verify that
(i) (A) B) x C = (AxC)0(BxC) . (ii) AX (B-C) = (A x B) - (AXC)
4 Relations​

Answer 1

D=the set of all natural numbers less than

Answer 2

Correct Question:

Let A = The set of all natural numbers less than 8, B = The set of all prime numbers less than 8, C = The set of even prime numbers

Which of the statement is true?

(i) $(A+B) \times C = (A \times C)$

(ii) $(B \times C) = 2$

(ii) $A\times (B-C) \ne (A \times B) - (A\times C)$

(iv) None of them

(v) All of them

Final Answer:

When A = The set of all natural numbers less than 8, B = The set of all prime numbers less than 8, C = The set of even prime numbers; the true statement is (v) All of them; as $(A+B) \times C = (A \times C)$,  $(B \times C) = 2$, and $A\times (B-C) \ne (A \times B) - (A\times C)$ are all true statements.

Given:

Let A = The set of all natural numbers less than 8, B = The set of all prime numbers less than 8, C = The set of even prime numbers.

To Find:

The true statement among the following.

(i) $(A+B) \times C = (A \times C)$

(ii) $(B \times C) = 2$

(ii) $A\times (B-C) \ne (A \times B) - (A\times C)$

(iv) None of them

(v) All of them

Explanation:

Note these vital points.

• A set is a pre-defined group of elements such that they share a common attribute among them.
• The addition of any two sets indicates all the elements present in either of the sets.
• The difference between any two sets indicates all the elements obtained after removing al the elements of the second set from the first set.
• The multiplication of any two sets indicates those elements only that are present in each of the sets.

Step 1 of 5

As per the given statements in the  problem, write the following.

• A = The set of all natural numbers less than 8 = {1, 2, 3, 4, 5, 6, 7}.
• B = The set of all prime numbers less than 8 = {2, 3, 5, 7}.
• C = The set of even prime number = {2}.

Step 2 of 5

In accordance with the statement in the given problem, derive the following.

• The elements in $(A+B)$ are the elements in either of A or B = {1, 2, 3, 4, 5, 6, 7}.
• The elements in $(A+B) \times C$ are the elements in (A+B) and C = {2}.
• The elements in $(A \times C)$ are the elements in A and C = {2}.
• Hence, the statement $(A+B) \times C = (A \times C)$ is true.

Step 3 of 5

Again, from the statement in the given problem, deduce the following equation.

• The elements in $(B \times C)$ are the elements in B and C = {2}.
• Hence, the statement $(B \times C) = 2$ is true.

Step 4 of 5

In accordance with the statement in the given problem, observe the following.

• The elements in $(B-C)$ are the elements obtained after removing all the elements of the second set C from the first set B = {1, 4, 6}.
• The elements in $A\times (B-C)$ are the elements in A and (B-C)  = {1, 4, 6}.
• The elements in (A x B) are the elements in A and C = {2}.
• The elements in (A x C) are the elements in A and C = {2}.
• The elements in (A x B) - (A x C) are = { } or 0 or null set.
• Hence, the statement  $A\times (B-C) \ne (A \times B) - (A\times C)$ is true.

Step 5 of 5

Now, from the above calculations, it is clear that $(A+B) \times C = (A \times C)$, $(B \times C) = 2$, and $A\times (B-C) \ne (A \times B) - (A\times C)$ are true. Hence the correct option is (v) All of them.

Therefore, the required true statement is (v) All of them as$(A+B) \times C = (A \times C)$,  $(B \times C) = 2$, and $A\times (B-C) \ne (A \times B) - (A\times C)$ are all true statements, for A = The set of all natural numbers less than 8, B = The set of all prime numbers less than 8, and C = The set of even prime numbers.

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