let's find the letters in the word
'mathematics':
A = {m, a, t, h, e, i, c, s)
Now, let's use these sets A and B to verify
the given identities:
(i) Au(AnB) = An(AUB)
We need to prove that the left-hand side (LHS) and the right-hand side (RHS) of the identity are equal.
LHS: Au(AnB)
= {m, a, t, h, e, i, c, s) u ((a, t, i} n (v, e, r, i, f, y})
= {m, a, t, h, e, i, c, s) u (a, i}
= {m, a, t, h, e, i, c, s}
RHSAn(AUB)
=\ \{m, a, t, h, e, i, c, s\} * n \ m,a,t,h,e,i,c,s,v e, r, i, f, y)) u ([m, at, h, e, i, c, s] n (ve, r, i, f, y ])
= \{m, a, t, h, e, i, c, s\} * u\{a, i\}
= \{m, a, t, h, e, i, c, s\}
Since LHS RHSthe identity is verified
=
(ii) A - (AB) =(A cup B)-B
Again, we need to prove that the LHS and RHS of the identity are equal
LHS: A-(AB)
= {m, a, t, h, e, ic, s) - ([m, a, t, h, e, ic, s} n
\{v, e, r, i, f, y\} )
= \{m, a, t, h, e, i, c, s\} - \{i, e\}
= \{m, a, t, h, c, s\}
RHS: (AUB) - B
=([m, a, t, h, e, i, c, s) u (v, e, r, i, f, y]) - {v, e, r,i, f, y}
= {m, a, t, h, e, i, c, s]
= {m, a, t, h, c, s}
Since LHS = RHS, the identity is verified
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Answers & Comments
let's find the letters in the word
'mathematics':
A = {m, a, t, h, e, i, c, s)
Now, let's use these sets A and B to verify
the given identities:
(i) Au(AnB) = An(AUB)
We need to prove that the left-hand side (LHS) and the right-hand side (RHS) of the identity are equal.
LHS: Au(AnB)
= {m, a, t, h, e, i, c, s) u ((a, t, i} n (v, e, r, i, f, y})
= {m, a, t, h, e, i, c, s) u (a, i}
= {m, a, t, h, e, i, c, s}
RHSAn(AUB)
=\ \{m, a, t, h, e, i, c, s\} * n \ m,a,t,h,e,i,c,s,v e, r, i, f, y)) u ([m, at, h, e, i, c, s] n (ve, r, i, f, y ])
= \{m, a, t, h, e, i, c, s\} * u\{a, i\}
= \{m, a, t, h, e, i, c, s\}
Since LHS RHSthe identity is verified
=
(ii) A - (AB) =(A cup B)-B
Again, we need to prove that the LHS and RHS of the identity are equal
LHS: A-(AB)
= {m, a, t, h, e, ic, s) - ([m, a, t, h, e, ic, s} n
\{v, e, r, i, f, y\} )
= \{m, a, t, h, e, i, c, s\} - \{i, e\}
= \{m, a, t, h, c, s\}
RHS: (AUB) - B
=([m, a, t, h, e, i, c, s) u (v, e, r, i, f, y]) - {v, e, r,i, f, y}
= {m, a, t, h, e, i, c, s]
= {m, a, t, h, c, s}
Since LHS = RHS, the identity is verified