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Set Theory

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Q 1. {tex}N{/tex} is the set of natural numbers. The relation R is defined on {tex} N \times N {/tex} as follows: {tex} \left(a,\ b\right)R \left(c,\ d\right) \Leftrightarrow a+d=b+c {/tex} is:

reflexive

symmetric

transitive

all of these

(a) {tex} \left(a,\ b\right) R \left(a,\ b\right) \Leftrightarrow \ a+b=b+a {/tex}

Therefore, R is reflexive

(b) {tex} \left(a,\ b\right) R \left(c,\ d\right) => a+d = b+c {/tex}

=> {tex} c+b = d+a {/tex}

=> {tex} \left(c,\ d\right) R \left(a,\ b\right) {/tex}

Therefore, R is symmetric

(c) {tex} \left(a,\ b \right) R \left(c,\ d\right)\ and\ \left(c,\ d\right) R \left(e,\ f \right) {/tex}

=> {tex} a+d=b+c\ and \ c + f=d+e {/tex}

=> {tex} a+b+c+f = b+c+d+e {/tex}

=> {tex} a+f = b+e {/tex}

=> {tex} \left(a,\ b\right) R \left(e,\ f\right) {/tex}

Therefore, R is transitive

Thus R is an equivalence relation {tex} N \times N {/tex}

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Q 2. {tex} A=\{ 4, 9, 16, 25, 36 \},\ B=\{ 9, 25, 49, 81 \},\ C=\{ 16, 81, 256 \} {/tex} then find {tex} A- \left(B\cap C\right) {/tex}

{tex} \{ 81 \} {/tex}

{tex} \{ 4, 9, 16, 25, 36 \} {/tex}

{tex} \{ 4, 36 \} {/tex}

{tex} \{ 4, 16, 36 \} {/tex}

{tex} A=\{ 4,\ 9,\ 16,\ 25,\ 36 \},\ B=\{ 9,\ 25,\ 49,\ 81 \},\ C=\{ 16,\ 81,\ 256 \} {/tex}

{tex} B\cap C = \{ 81 \} {/tex}

{tex} A- \left(B\cap C\right) = \{ 4,\ 9,\ 16,\ 25,\ 36 \}-\{ 81 \} {/tex}

={tex} \{ 4,\ 9,\ 16,\ 25,\ 36 \} {/tex}

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