Q1
Which of the following sets have closure property w.r.t addition and multiplication?

i. {0}

SOLUTION:

BY ADDITION: 0+0=0 ∈ {0}

BY MULTIPLICATION: 0x0=0 ∈ {0}

Hence {0} has closure property w.r.t addition and multipication.

(ii) {1}

SOLUTION:

BY ADDITION: 1+1=2 ∉ {1} => {1} does not have closure property w.r.t addition.

BY MULTIPLICATION: 1x1=1 ∈ {1} => {1} has closure property w.r.t multiplication.

(iii) {0,-1}

SOLUTION:

BY ADDITION: 0+0=0 ∈ {0,-1} and (-1)+(-1)= -1 -1= -2 ∉ {0,-1} =>{0,-1} doest not have closure property w.r.t addition

BY MULTIPLICATION: 0x0=0 ∈ {0,-1} and -1 x -1 = 1 =>{0,-1} doest not have closure property w.r.t multiplication.

(iv) {1,-1}

SOLUTION

BY ADDITION: 1 + 1 = 2 ∉ {1,-1} and (-1) + (-1) = -1-1 = -2 ∉ {1,-1}

=>{1,-1} doest not have closure property w.r.t addition

BY MULTIPLICATION: 1 x 1 = 1 ∈ {1,-1} and -1 x -1 = 1 ∉ {1,-1}

=>{1,-1} doest not have closure property w.r.t multiplication.