Question

If A={a,b,c,d,e} , B={a,c,e,g} and C={b,e,f,g}, then verify that

i)A∩(B-C)=(A∩B)-(A∩C)

ii)A-(B∩C)=(A-B)∪(A-C)

will mark the answer as brainliest for sure ​

Answer 1

$\large\bf\underline {To \: find:-}$

• we need to verify that
• i) A ∩(B-C) = (A ∩ B) - (A ∩ C)
• ii) A - (B ∩ C) = (A - B)∪(A - C)

$\large\bf\underline{Given:-}$

• A = {a,b,c,d,e}
• B = {a,c,e,g}
• C = {b,e,f,g},

$\huge\bf\underline{Solution:-}$

A , B and C are three sets as :-

• A = {a, b ,c, d, e}
• B = {a, c, e, g}
• C = {b ,e ,f, g}

i) A ∩(B - C) = (A ∩ B) - (A ∩ C)

➛ (B - C) = { g }

➛ A ∩(B - C) = { }

➛ (A ∩ B) = { a ,c , e }

➛ (A ∩ C) = { b, e, }

➛ (A ∩ B) - (A ∩ C) = { a , c }

Now,

⚘ Verification :-

➛ A ∩(B - C) = (A ∩ B) - (A ∩ C)

➛ ᛄ ≠ {a ,c}

So,

LHS ≠ RHS

ii) A - (B ∩ C) = (A - B)∪(A - C)

• A = {a, b ,c, d, e}
• B = {a, c, e, g}
• C = {b ,e ,f, g}

➛ (B ∩ C) = { e , g }

➛ A - (B ∩ C) = { a ,b ,c ,d }

➛ (A - B) = { b ,d }

➛ (A - C) = { a ,c ,d }

➛ (A - B) ∪ (A - C) = { a ,b , c ,d }

⚘ Verification :-

➛ A - (B ∩ C) = (A - B)∪(A - C)

➛ {a , b, c ,d } = { a ,b ,c ,d }

LHS = RHS

Hence,

• ❥ A ∩(B-C) ≠ (A ∩ B) - (A ∩ C)
• ❥ A - (B ∩ C) = (A - B)∪(A - C)

━━━━━━━━━━━━━━━━━━━━━━━━━

Answer 2

$\bold\purple{\underline{\bf {Given:-}}}$

• A = {a, b, c, d, e}
• B = {a, c, e, g}
• C = {b, e, f, g}

$\bold\pink{\bf{\underline{To\:Find:-}}}$

i) A ᑎ (B - C) = (A ᑎ B) - (A ᑎ C)

ii) A - (B ᑎ C) = (A - B) ᑌ (A - C)

$\huge\bold\red{\bf{\underline{Solution:-}}}$

(i)

★ (B - C)

= {a, c, e, g} - {b, e, f, g}

= {a, c}

★ (A ᑎ B)

= {a, b, c, d, e} ᑎ {a, c, e, g}

= {a, c, e}

★ (A ᑎ C)

= {a, b, c, d, e} ᑎ {b, e, f, g}

= {b, e}

Left Side :-

A ᑎ (B - C)

= {a, b, c, d, e} ᑎ {a, c}

= {a, c}

Right Side :-

(A ᑎ B) - (A ᑎ C)

= {a, c, e} - {b, e}

= {a, c}

∵Left Side = Right Side

∴A ᑎ (B - C) = (A ᑎ B) - (A ᑎ C) [Hence Verified]

___________________________

(ii)

★ (B ᑎ C)

= {a, c, e, g} ᑎ {b, e, f, g}

= {e, g}

★ (A - B)

= {a, b, c, d, e} - {a, c, e, g}

= {a, b, d}

★ (A - C)

= {a, b, c, d, e} - {b, e, f, g}

= {a, c, d}

Left Side :-

A - (B ᑎ C)

= {a, b, c, d, e} - {e, g}

= {a, b, c, d}

Right Side :-

(A - B) ᑌ (A - C)

= {a, b, d} ᑌ {a, c, d}

= {a, b, c, d}

∵ Left Side = Right Side

∴ A - (B ᑎ C) = (A - B) ᑌ (A - C) [Hence Verified]